a model of consumer brand and quantity choice dynamics under
TRANSCRIPT
Brand and Quantity Choice Dynamics under Price Uncertainty
Tülin Erdem
University of California at Berkeley
Susumu Imai
Concordia University
Michael P. Keane
Yale University
December 2001
Revised, September 2002
Note: An earlier (June 1998) draft of this paper was presented under the title “Consumer Price and
Promotion Expectations: Capturing Consumer Brand and Quantity Choice Dynamics under Price
Uncertainty.” The authors thank seminar participants at AGSM, Hebrew University, NYU, MIT, Harvard
and Berkeley for useful comments on the earlier draft. We also thank seminar participants at the 2002
North American Winter Meetings of the Econometric Society in Atlanta and the University of Chicago
for valuable comments on the later version. This research has been funded by NSF grant SBR-9511280.
Abstract
We develop a model of household demand for frequently purchased consumer goods that
are branded, storable and subject to stochastic price fluctuations. Our framework accounts for
how inventories and expectations of future prices affect current period purchase decisions. We
estimate our model using scanner data for the ketchup category. Our results indicate that price
expectations and the nature of the price process have important effects on demand elasticities.
Long-run cross price elasticities of demand are more than twice as great as short-run cross price
elasticities. Temporary price cuts (or “deals”) primarily generate purchase acceleration and
category expansion, rather than brand switching.
Keywords: price expectations, pricing, scanner data, dynamic programming, simulation, discrete
choice.
1
The goal of this paper is to develop and estimate a dynamic model of consumer choice
behavior in markets for goods that are: 1) frequently purchased, 2) branded, 3) storable, and 4)
subject to frequent price promotions, or “deals.” In such an environment, forward-looking
behavior of consumers is important. Specifically, optimal purchase decisions will depend not
only on current prices and inventories, but also on expectations of future prices. There is no
single “price elasticity of demand.” Rather, the effect of price changes on consumer demand will
depend upon how the price change effects expectations of future prices. This depends on the
extent to which consumers perceive the price change to be permanent or transitory, and the
extent to which they expect competitor reaction. These, in turn, depend on the stochastic process
for prices in the market (see Marshak (1952), Lucas (1976)).
In recent years a wealth of supermarket scanner data have become available that
document sales of frequently purchased consumer goods. In a number of instances, panels of
households have been provided with individual ID cards, so that all their purchases over long
periods of time can be tracked. These data provide a valuable opportunity to study consumer
choice dynamics. We will argue that such analysis is important not only for marketers wishing to
predict consumer response to promotions, but also for economists interested in firm pricing
behavior, antitrust policy, welfare gains from introduction of new goods, construction of price
indexes, etc.
Since the pioneering work of Guadagni and Little (1983), an extensive literature has
emerged that uses scanner data to study consumer choice behavior. But for the most part, this
literature has relied on static models of consumer behavior, in the sense that consumers make
decisions to maximize current period utility. Much of this literature has dealt with the issue of
choice “dynamics,” where dynamics is used to refer to purchase carry over effects (or habit
persistence) – i.e., does past purchase of a brand increase a consumer’s current period utility
from purchase of that brand (see, e.g., Keane (1997a)). But none of the published literature
examines consumer choice “dynamics” in the sense of how expectations of future prices
influence the current period purchase decisions of forward looking consumers.1
1 Erdem and Keane (1996) develop a model of forward looking consumers, but the focus there is on learning about brand quality in an environment where consumers have uncertainty about brand attributes. This generates a motive for trial or experimental purchases of brands to facilitate learning. Erdem and Keane model prices as iid over time, so changes in current prices do not alter expected future prices.
2
Understanding the role of price expectations in consumer purchase behavior is important
for many reasons. For instance, evaluations of the welfare effects of mergers and the welfare
gains from introduction of new goods (see Hausman (1997)), rely on estimates of own and cross-
price elasticities of demand for the goods in question. But the existing literature only contains
static elasticity estimates. To our knowledge, there is no published work that reports dynamic
price elasticities of demand that take into account how a price cut today affects consumers
expectations of prices tomorrow, or that distinguishes among elasticties of demand with respect
to price cuts that are perceived to have different degrees of persistence.2 We provide a
framework for estimating dynamic price elasticities of demand for branded frequently purchased
consumer goods. We will show that accounting for dynamics can have large effects on own and
cross-price elasticity estimates.
More generally, our framework can be used to predict how consumers’ purchase decision
rules would respond to changes in the entire retail pricing process (such as, for example, a shift
from high/low (H/L) pricing to “everyday low pricing” or EDLP). To our knowledge there is no
prior structural work that enables one to predict consumer response to “major” pricing policy
changes.3 This problem is apparently understood by marketing practitioners. For example, in a
criticism of existing models of promotion response Struse (1987), a marketing manager at
General Mills, observed that: “To make a significant contribution to management practice, these
models … need to focus on the forecasting of future promotions effectiveness. While analysis of
past events may be … useful, the real need is to better predict the future - especially under
interesting circumstances. That is, the manager needs a forecasting method which will be robust
and discriminating over a wider range of conditions than actually seen in the market since he or
she needs to explore alternatives which go beyond past practice …”
Understanding consumers’ dynamic responses to pricing policy changes may also be
important for understanding industry dynamics. Existing dynamic oligopoly models that
2 In “market mapping” methods (see Elrod (1988)) cross-price elasticities of demand are critical for the evaluation of the positioning of products in unobserved (or latent) attribute space. Srinivasan and Winer (1994) and Erdem (1996) discuss how “dynamics” in the sense of habit persistence may distort such evaluations. How dynamics in the sense of price expectation formation might distort such evaluations has not been considered. 3 See Keane (1997b) for a discussion of this issue. To give an example of the problem, we would expect that price elasticities of demand would differ between an EDLP regime and a H/L regime for a variety of reasons. For instance, a price cut has different effects on expected future prices under each regime, and the expected duration until the next price cut is different under each regime. As a result, one can’t use estimates obtained under the H/L regime to predict behavior under the EDLP regime, unless one uses a structural model like ours.
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endogenize price (see, e.g., Berry, Levinson and Pakes (1995)) typically assume that consumer
behavior is static. This may be a serious misspecification in markets where purchases are made
frequently, and changes in current prices lead to important changes in expected future prices. We
hope that work like ours will eventually lead to elaboration of the consumer side of dynamic
oligopoly models.4
Understanding how forward looking consumers respond to temporary price cuts is
important for retailers and brand managers, who want to know if price cuts merely cause
consumers to accelerate purchases, or whether they also induce brand switching and/or increased
category sales. Furthermore, the design of intertemporal price discrimination strategies requires
an understanding of how changes in the whole price process affects consumer demand (e.g.,
would more frequent promotion generate sales to new consumers, or simply alter the purchase
timing of existing consumers?).
As a final example, an understanding of the dynamics of consumer purchase behavior is
important for the construction of price indices. To some extent, this involves the random
sampling of posted supermarket prices, which will capture average offer prices.5 But, if a large
share of purchases occurs on promotion, then the average offer price of a good is not the relevant
measure of its typical cost to consumers. In fact, a widespread shift from H/L pricing to EDLP,
such as occurred in the US in the late 80’s and early 90’s (see Lal and Rao (1997) for a
discussion), could cause the average posted price to fall even though the average purchase price
does not, thus distorting price level estimates based on random sampling of posted prices. Our
framework allows one to estimate the relationship between mean offer and accepted prices under
alternative price processes.
4 The computational capacity and econometric methods needed to estimate equilibrium models with forward looking behavior on both the firm and consumer side are probably several years away. But we should note that Ching (2002) has estimated a model of the pharmaceutical industry with dynamics on both sides of the market in stages. First, the demand side model is estimated jointly with an approximate reduced form equation for firm’s pricing policy function. In a second stage the remaining supply side parameters are calibrated, treating the demand side parameters as known. 5 The BLS website (see www.bls.gov/cpi/cpifact2.htm) contains some description of the random sampling that underlies construction of the CPI. According to their description, “… during the first week of the month … BLS data collectors (called economic assistants) gather price information from selected department stores, supermarkets, service stations, doctors' offices, rental units, etc. For the entire month, about 80,000 prices are recorded in 87 urban areas. During each monthly visit, the economic assistant collects price data on a specific good or service with precisely defined qualities or characteristics. If the selected item is available, the assistant records its price…. The Bureau does not have resources to price every good or service in every retail outlet in every urban area of the country. Therefore, a scientifically selected sample must be used to make the CPI representative of the prices paid for all goods and services purchased by consumers in all urban areas.”
4
In this paper, we estimate our model of consumer brand and quantity choice dynamics on
scanner panel data provided by A.C. Nielsen. We use the data on household ketchup purchases.
We choose the ketchup category for two reasons. First, it satisfies the four criteria discussed at
the outset.6 Second, of the goods that satisfy these criteria and for which scanner data have been
released for public use, ketchup is the easiest category to work with. This is because the number
of brand/size combinations for ketchup is lower than for the other available categories (there are
4 brands – Heinz, Hunts, Del Monte and the Store brand – that come in 3 to 5 sizes each, giving
a choice set with 16 elements). We felt it was sensible to first apply our framework to this
category before tackling categories with more brands and/or sizes (such as yogurt, toilet paper,
cereal, etc.).
Our estimated model provides a remarkably good fit to all the important dimensions of
the data, including brand shares, size shares, purchase frequency, inter-purchase times, purchase
hazard rates, brand switching matrices, and the distributions of accepted prices.
We use simulations of the model to evaluate the importance of price expectations. For
instance, we can simulate the effect of a temporary price cut for one brand, both allowing for the
effect of this price cut on expected future prices, and holding expectations fixed. Since the price
process for ketchup exhibits substantial persistence, we find, as one would expect, that the
current period increase of own brand sales in response to a temporary price cut is dampened by
the expectations effect. However, this dampening effect is rather modest. For example, it is about
10% for the leading brand - Heinz. Interestingly, however, we find that the cross-price effects
that account for expectations are roughly twice as large as cross-price effects holding
expectations fixed. For example, the percentage drop in current period sales for Hunts, Del
Monte and the Store brand are roughly twice as great if we account for the effect of the Heinz
price cut on expected future prices of all the brands.
Two factors drive this key result: 1) if Heinz’ price is lowered today it leads consumers to
also expect a lower Heinz price tomorrow. This lowers the value function associated with
6 Pesendorfer (2002) reports that there is little evidence of seasonality in ketchup prices, and that cost factors seem unrelated to short run price movements. He argues that the frequent week to week price fluctuations of ketchup are most plausibly explained by a type of inter-temporal price discrimination strategy on the part of firms, in which the retailers play mixed strategies. We agree with this analysis, which supports the view that price movements are exogenous from the point of view of consumers. We believe that similar factors are at work in most frequently purchased consumer goods markets. It is highly implausible that the high frequency price movements observed in these markets (e.g., price cuts that last only a few days or a week) could be due to demand shocks that are observed by firms.
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purchase of any brand other than Heinz today. 2) Given the price dynamics in the ketchup
market, a lower price of Heinz today leads consumers to expect competitor reaction, so it lowers
the expected prices of the other brands tomorrow. This further lowers the value associated with
purchase of those brands today.
Obviously, the quantitative significance of these two effects depends on the price process.
Thus, a key point is that cross-price elasticities do not (by themselves) reveal the similarity of
differentiated products in attribute space (or their degree of competition). The magnitudes of
cross-price elasticities also depend on the price process - because this determines how a price cut
for one brand today affects expected prices of all brands in the future. Given the importance of
cross-price elasticities of demand in such areas as the analysis of mergers and the valuation of
new goods, our results clearly show that accounting for consumer price expectations may be
critical in these areas.
1. Background and Literature Review
Research on joint modeling of consumer brand and quantity decisions has a long tradition
in both marketing and economics. Hanneman (1984) developed a unified framework for
formulating econometric models of discrete (e.g., brand choice) and continuous choices (e.g.,
quantity decisions) in which the discrete and continuous choices both flow from the same
underlying utility maximization decision.7 Dubin and McFadden (1984) used such a model to
analyze residential electric appliance holdings and consumption. In marketing, Chiang (1991)
and Chintagunta (1993) also adopted the Hanneman framework and calibrated static models
consistent with random utility maximization on scanner panel data.
All these models assume that consumers are myopic in that they maximize immediate
utility. However, frequently purchased consumer goods typically exhibit substantial inter-
temporal price variation, which suggests that for storable goods consumer expectations about
future prices may play an important role in purchase timing and quantity decisions. Indeed, the
evidence of forward-looking behavior in frequently purchased consumer goods markets is
7 In Hanneman’s framework, the commonly observed phenomenon that consumers rarely (if ever) buy multiple brands of a frequently purchased product on a single shopping occasion is shown to arise if the brands are perfect substitutes, quantity is infinitely divisible and pricing is linear. In that case, the brand and quantity decisions separate: In stage 1 it is optimal to choose the brand with the highest utility per unit, and in stage 2 the consumer chooses the number of units conditional on that brand. Keane (1997b) pointed out that this separation does not go through if available quantities are discrete, as is the case with the large majority of frequently purchased consumer goods. However, the literature typically ignores this problem, and assumes quantity is continuous, because of the computational difficulty involved in modeling choice among a multitude of discrete brand/size combinations.
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overwhelming. For example, in descriptive analyses, Both Hendel and Nevo (2001a) and
Pesendorfer (2002) find that, conditional on current price, current demand is higher when past
prices were higher or time since last sale is longer (implying that past sales were lower, and
hence that current inventories are lower). This implies that consumers “stock up” on storable
goods when they see a “deal.”
Shoemaker (1979) and Ward and Davis (1978) were perhaps the first (of many) studies to
find evidence of “purchase acceleration,” meaning that deals induce consumers to buy larger
than normal quantities. Neslin, Henderson and Quelch (1985) found that advertised price cuts led
to both shorter interpurchase time and larger purchase quantities for coffee. Hendel and Nevo
(2001a) confirm this for 3 more products, and also find that duration to next purchase is longer
following a deal purchase. It is the combination of both increased current purchases and longer
duration to next purchase that one needs forward-looking behavior to explain. While a static
model with an outside good can explain a current increase in category sales in response to a
temporary price cut, the increase in duration to next purchase implies that consumers time
purchases to coincide with prices that are “low” relative to some inter-temporal standard.
The large literature on “reference prices,” starting with Winer (1986), consistently finds
that consumers base current purchase decisions not just on current prices but also on how these
relate to some inter-temporal pricing standard, which may be defined as an average or typical
price for the product. This is highly suggestive that expectations of future prices affect consumer
purchase decisions.8
There is also clear (recent) evidence that the Lucas Critique is quantitatively relevant.
Mela, Jedidi and Bowman (1998) examine 8 years of data for a frequently purchased consumer
product. During the last 6 quarters of their data there was a regime shift where deals became
much more frequent. Under the new regime: 1) consumers bought less often, concentrating their
purchases in deal periods, 2) consumers bought larger quantities when they did buy, and 3)
overall sales were roughly constant. Mela et al conclude that, under the new regime, consumers
“learned to lie in wait for deals.” Furthermore, Kopalle, Mela and Marsh (1999) find (for several
products) that increased frequency of promotion reduces “baseline sales” of a brand, and also
increases its price elasticity of demand. 8 For instance, Winer posited that the “reference price” reflects the expectations of the consumer, which are shaped by the past pricing activity of the brand. However, in the marketing literature, psychological explanations have also been offered for this type of finding.
7
The behavior of retail prices also provides indirect evidence for the importance of
forward-looking behavior by consumers. Both Pesendorfer (2002) and Hong, McAfee and
Nayyar (2002) point out that it is hard to explain observed serial correlation in retail prices
without consumer stockpiling behavior. In static price discrimination story, a la Varian (1980),
prices should be iid over time. In contrast, suppose there exists a segment of price sensitive
consumers who stockpile the good and “lie in wait for deals,” creating scope for intertemporal
price discrimination. As time since the last sale increases, the number of price sensitive
consumers looking to buy grows, which increases potential revenue from a sale. Eventually, the
retailer decides to have a sale, and then quickly returns price to the “regular” level. This positive
duration dependence in the probability of a deal is in fact the price pattern observed for
frequently purchased storable consumer goods.
In the marketing literature there are two influential papers that examined the purchase
timing, brand choice and quantity decision of consumers for frequently purchased storable
consumer goods. These are Gupta (1988) and Chintagunta (1993). Gupta models all three
decisions, but the decisions are not linked, and there is no consumer taste heterogeneity.
Chintagunta models all three choices in a unified utility maximization framework, and he allows
for consumer taste heterogeneity. Interestingly, these two papers reach opposite conclusions
regarding a key issue: Gupta concludes that most increased sales from a temporary price cut are
due to brand switching, and that cross-price elasticities of demand are large. In contrast,
Chintagunta finds that most increased sales from a temporary price cut are due to purchase
acceleration by brand loyal consumers, and concludes that cross-price elasticities of demand are
small. The Gupta results are the main evidence in the literature that is taken as unfavorable for
dynamics/stockpiling behavior.
In fact, the contrast between the Gupta (1988) and Chintagunta (1993) results is exactly
what one would expect if forward-looking/stockpiling behavior is important. The difference in
results would then be generated by dynamic selection and endogeneity bias. To see this, consider
the following example. Suppose Brand A has a deal in period t. Then, the population of people
who buy the category at t has an over representation of people “loyal” to A. In a static logit brand
choice model, such as in Gupta (1988), low price for a brand is therefore correlated with high
taste for the brand. As a result, cross-price effects are overestimated. Chintagunta (1993) deals
with this selection bias because he allows for taste heterogeneity. Indeed, Sun, Neslin and
8
Srinivasan (2001) show, using simulations, that static choice models without heterogeneity
drastically overstate cross-price elasticities if consumers engage in stockpiling behavior. Recently, there have been a number of papers dealing with the issue of potential
endogeneity of prices in consumer choice models (see, e.g., Nevo (2001)). In our view, much of
this literature has missed the mark, because it has failed to make a crucial distinction between
endogeneity stemming from aggregate (market) demand shocks and endogeneity stemming from
omitted variables. Frequently purchased consumer goods typically exhibit price patterns in which
prices stay flat for weeks or months at a time (“regular price”), and then exhibit short-lived drops
(“deals”). We find it extremely implausible that these deals are the result of manufacturer,
wholesaler or retailer responses to aggregate taste shocks, for several reasons. Why would
demand for a good like ketchup or yogurt suddenly jump every several weeks and then return to
normal? And how could sellers detect such a jump quickly enough to incorporate it into daily or
weekly price setting? Thus, we feel the most plausible explanation for the observed price
variation is some sort of inter-temporal price discrimination, such as that considered by
Pesendorfer (2002) and/or McAfee and Nayyar (2002).
On the other hand, an important reason for endogeneity of prices in demand models is the
failure to account for consumer inventories, which are not observed in scanner data. If prices are
persistent over time and consumers engage in stockpiling behavior, then inventories will be
correlated with current prices. This causes price to be econometrically endogenous due to the
omitted variables problem, even though price fluctuations are exogenous from the point of view
of the consumer.9 The correct way to deal with this endogeneity problem is to estimate a model
of dynamic choice behavior, and to integrate out the unobserved latent inventory levels from the
likelihood function. This is extremely computationally demanding, but it is exactly what we do
in this paper.10
To our knowledge there is no published research that structurally estimates a model of
consumer brand and quantity choice dynamics for frequently purchased storable consumer goods
under price uncertainty. After our work on this project was largely complete we became aware of
ongoing work by Hendel and Nevo (2001b), who are attempting to estimate a structural model
9 We thank Steve Berry for pointing this out to us. 10 Of course, an alternative would be a BLP type procedure using instruments for price that are uncorrelated with the unobserved inventories. But such instruments would be difficult to find.
9
that is in some ways similar to ours. In the course of presenting our model (in the next section)
we will provide some discussion of how their approach differs from ours.
2. The Model
2.1. Overview
In our model, the good is storable, and households get utility from its consumption.
Brands differ in the utility they provide per unit consumed. A key aspect of the model is that
consumers have a per period usage requirement for the good, which is stochastic, and which is
only revealed after the purchase decision is made. Thus, households run a risk of stocking out of
the good if they maintain an inadequate inventory to meet the usage requirement. There is a cost
of stocking out. At the same time, there are carrying costs of holding inventories, and fixed costs
of making purchases. The prices of each brand evolve stochastically according to a (vector)
stochastic process that is known to consumers.11
The model incorporates consumer heterogeneity in two ways: First, we allow for four
types of consumers in terms of their vector of utility evaluations for the brands. Second, we also
allow for four types of consumers in terms of the usage rate. Thus, there are sixteen types in all.
We find that this degree of heterogeneity allows us to fit the data very well. A novel aspect of
our model is that a household’s usage rate type evolves over time according to a Markov process.
A salient feature of the data is that households will often be frequent purchasers of ketchup for
several months, then stop buying ketchup for several months, etc. Allowing usage rate type to
evolve stochastically over time allows us to capture this type of pattern.
A vital component of our model is the price process, which we estimate separately in a
first stage, using the price data from Nielsen. We estimate a multivariate jump process that
captures three key features of the data: 1) prices typically are constant for several weeks,
followed by jumps, 2) the probability and direction of jumps depends on competitor prices, and
11 Rather than assuming that households have a usage requirement, we could have instead assumed that utility is concave in consumption. It is worth highlighting how the model’s behavior would differ in these two cases. In each case, a permanent price increase would reduce consumption and therefore sales. In our model households reduce consumption by suffering more frequent stock-outs (while continuing to consume at the same rate when the good is in stock). In the alternative model with concave utility, households reduce consumption by slowing down their rate of consumption. Since we don’t observe actual consumption in the data, we don’t feel this distinction is likely to be important. We adopt our formulation because it is a bit simpler. Also, for many of the types of goods we are interested in, such as ketchup, toilet paper, laundry detergent, etc., we think that consumers have a satiation point beyond which they do not derive additional utility from added consumption (e.g., you don’t get extra utility from using more detergent beyond what is needed to do your laundry, or using more ketchup beyond what the kids want on their hamburgers). Ailawadi and Neslin (1998) provide evidence consistent with this view.
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3) the direction of jumps depends on own lagged price (so the jump process is autoregressive).
This stochastic model is innovative in itself – we are aware of no prior modeling that actually
provides a good fit to the rather complex price processes observed in supermarket scanner data.
2.2. Household Utility
We assume that households have utility functions defined over consumption of each
brand of a particular good and a composite other commodity. Denote the per period utility
function for household i at time t by:
),...,,( 1 itiJttiit ZCCUU �
where Cijt is the quantity of brand j consumed by household i at time t, and Zit is the quantity of
the outside good that is consumed. Utility depends on quantities consumed rather than quantities
purchased because the good in question is storable and households hold inventories. To simplify
the model we assume that the composite good is not storable.
Further, we assume that utility is linear in consumption and additively separable between
the storable commodity and the composite other good, so Uit takes the form:
(1) �
�
��
Jjitijtijit ZCU
,1�
where �ij represents household i's evaluation of the efficiency units of consumption provided by
each unit of brand j. The assumption of perfect substitutability among brands, and that brands
generate differential utility per unit consumed, is similar to the set up in Hanneman (1984). This
linear form allows us to ignore saving decisions, so that the only inter-temporal link in the model
comes through inventories. We view this simplification as desirable, since the focus of our study
is on inventory decisions and not saving decisions.
We model unobserved heterogeneity in consumer evaluations of the efficiency units of
consumption, �ij, by adopting a finite mixture approach (e.g., Heckman and Singer (1984),
Kamakura and Russell (1989)). Thus, we assume that there are m types and we estimate type-
specific parameters for the evaluation of the efficiency units of consumption, �mj, along with the
probability that a household is type m, which we denote by � . As noted earlier, we found that a
model with m=4 gave an excellent fit to the data.
m
We assume that households can only purchase a single brand j on a given purchase
occasion t. This is consistent with the observation that for most frequently purchased consumer
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goods, households rarely if ever buy multiple brands on a single purchase occasion. For each
brand j, the household can choose among a discrete set of available quantities (which we will
enumerate in the data section).
The budget constraint for household i at time t is:
(2) � �������
jititititijtijtitijtijt YZSCCCQQDQP )( 2
321 ���
where Pijt is the price of brand j to household i at time t, Qijt is the quantity of j purchased by i at
t, and Yit is income of i at t. We enter terms for fixed cost of a purchase, inventory carrying cost,
and stockout cost in the budget constraint, although, since utility is linear in consumption, it is
irrelevant whether these enter here or in the utility function.
The term 21 �� � is the fixed cost associated with a purchase. This consists of
a constant term and a quadratic in the size purchased, and can be interpreted, for instance, as the
cost of going to the store, and then carrying the container home.12 The purchase indicator Dit=1
if a purchase is made and zero otherwise.
23 ijtijt QQ ��
The term CCit is the cost associated with carrying an inventory of the storable good
under analysis for household i during time period t. Finally, SCit is the fixed stock out cost
incurred by household i during time period t if their usage requirement exceeds their inventory.
We will further define CCit and SCit in Section 2.3.1 below. Substituting for Zit in (1) using (2)
we obtain:
(3) ititijtijtJj Jj
itijtijtitijtijit SCCCQQDQPYCU �������� � �� �
)( 2321
,1 ,1
����
Because Yit enters the conditional indirect utility function given purchase of each brand j in the
same way, Yit will not affect brand choice decisions and can be ignored in the model.13
12 Note: We had trouble generating the concentration of ketchup purchases at the most popular (32oz) size without the quadratic in size, which, in our estimates, implies a fixed cost that is convex in the amount purchased. We can generate that 32oz is clearly the most popular size even without this quadratic, but not to the same degree seen in the data. This same problem – the extent of preference for the most popular size is hard to explain - has been noted in many past marketing studies. These typically invoke size specific preferences (or “size loyalty”) to explain the phenomenon. Our convex cost holds down purchases of the larger (40oz, 64 oz) sizes. We speculate that the extent of preference for the 32 ounce size actually has a lot to do with the visibility of this size in stores, due to end-of-aisle and in-aisle displays. This is probably what our quadratic in size captures. See also footnote 38 on this point. 13 An interpretation of the fact that price enters the conditional indirect utility linearly is that the marginal utility of consumption of the outside good is constant over the small range of potential expenditures on the inside good, since these expenditures will be very small relative to Yit. This type of assumption is standard in marketing studies of
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2.3. Household Inventories
2.3.1. Preliminaries
We assume that households have an exogenous stochastic usage need for the storable
commodity in each period, given by Rit, and that they only get utility from consumption of the
good up to the level determined by the usage need, and not beyond that level. Define
��
�
Jjijtit CC
,1
Then,
.itit RC �
The inequality allows for the possibility of stock outs in which case consumption falls short of
the desired amount. We assume that Rit is not revealed until after the purchase decision is made
at the start of period t.
The assumption of an exogenous usage need is very reasonable in many categories where
consumption is unlikely to be a function of inventories. Indeed, previous work in marketing (e.g.,
Ailawadi and Neslin 1998) suggests that this assumption holds in ketchup (the category we will
study).14 It is worth emphasizing that this assumption does not mean consumption is independent
of price. If price is high for an extended period of time, the households in our model will reduce
consumption by suffering more frequent stock outs – as opposed to consuming any ketchup that
they have in stock at a slower rate.15 In other words, all adjustment of consumption to price is
along the extensive rather than the intensive margin.
Finally, we allow the distribution of the stochastic usage requirement to be heterogeneous
across consumers. Thus,
)
,(~log llti NR ��
demand for inexpensive consumer goods. It is exactly correct because we specify that utility is linear in demand for the outside good, but is still approximately correct under more general utility specifications, provided the inside good is inexpensive. 14 In other words, consumers do not put less ketchup on their hamburgers when their stock is low. Rather, they use some desired amount of ketchup until they stock out. We suspect that such behavior is typical of many other categories, such as laundry detergent, toilet paper, etc. On the other hand, in other categories, such as yogurt or cookies, consumption rates are presumably dependent on inventories. 15 In our view it is largely irrelevant whether we model households as reacting to price changes by altering consumption along the intensive and/or extensive margin, because the consumption information that would distinguish between these scenarios is not available in scanner data. Furthermore, adding a weekly continuous consumption decision would vastly increase the computational burden of solving the household’s optimization problem. Thus, we felt that ignoring the intensive margin was a sensible modeling choice.
13
where l=1, �, 4 and l denotes the usage type, where l=1 has the highest usage rate, whereas l=4
has the lowest usage rate. We assume that usage rate type is independent of preference type.
Furthermore, we assume that a household’s usage rate type may vary over time following a
Markov switching process. Let �ii denote the probability that a household remains type i from
one week to the next, and let � denote the transition probability from type i to type j. We
assume that:
ij
30.1 ii
ij�
�
�
� � ji . �
This says that if a household changes type, it is equally likely to change to any of the other types.
Let denote the initial probability of being type i. In order to conserve on parameters, we
assume that the initial probability is related to the family size (measured at the start of the panel)
in the following way:
i�
famsizef z 2 log log 101 �� ��
famsizef z loglog 202 �� ��
303 log log �� �
famsizef z loglog 404 �� ��
where is the family size. famsize
We also allow a stock out to carry a fixed cost. Denote by Iijt the inventory that
household i holds of brand j at the start of period t. The total inventory of all brands is given by:
(4) ��
�
Jjijtit II
,1
Thus, if household i purchases Qjt units at the start of t, its maximum consumption during period
t is Iit+Qit. Define
(5)it
itit
RQI
a)( �
�
If I[a < 1]=1 a stock out occurs, where I[ ] denotes an indicator function for the event within the
brackets.
The stock out cost to household i in period t has a constant component, as well as a
component proportional to the magnitude of a shortfall, and is given by:
14
(6) )(][)( 10 ititititititit CRICRsCRIsSC �����
where s0 is the fixed cost and s1 is the per unit cost.
We further assume that the cost of carrying inventory is given by:
(7) 221 ititit IcIcCC ��
where itI is the average inventory level during period t, which is given by:
(8) ]1[]2
[]1[]2
[ �
�
����� aIQIaaIRQII itititititit
and where c1 and c2 are linear and quadratic terms in the average inventory level. Note that the
construction of itI depends on whether or not a stock out occurs during the period. If there is no
stock out (a � 1), it is constructed assuming that usage is spread smoothly over the period. In the
event of a stock out (a < 1), it is constructed assuming that usage is at a constant rate prior to the
stock out, and that the stock is zero afterwards.
2.3.2. Evolution of Household Inventories
At any t, a household might potentially have a number of brands in its inventory. In that
case, we would need to model the order in which brands are consumed within a period. This
would lead to greatly increased complexity of our model, for little payoff. In most categories of
frequently purchased consumer goods, consumers almost never buy multiple brands on a single
shopping occasion, and brand “loyalty” is strong, so inventory holdings will not exhibit much
brand heterogeneity. So, to avoid having to model the order of consumption within a period in
those rare instances where it would be relevant, we assume that in period t, after the minimum
usage requirement Rit is realized, households use each brand in their inventory proportionately to
meet their usage needs.
The state of a household at time t includes their time t inventories of each brand. If there
are a large number of brands, this means that the state space for the consumer’s dynamic
optimization problem will grow large. However, under the assumption that brands are used
proportionately to meet the usage requirement, it turns out to be that the consumers’ state can be
characterized by just two variables, that is, their total inventory as given in equation (4) and their
quality-weighted inventory, which we define by
��
�
Jkiktikit II
,1
1 �
15
Recall from equation (1) that �ik represents household i's of evaluation the efficiency units of
consumption provided by each unit of brand j. This is why we call I1it the “quality” weighted
inventory.16
After purchasing Qijt units of brand j, the total stock of the storable good is Iit+ Qijt,
since households are assumed not to buy multiple brands in a given time period. Because of the
assumption that households use each brand proportionately to meet their usage needs, if the total
amount of the storable good is greater than or equal to the minimum usage requirement Rit, then
only a fraction 1/a of the stock of each brand is used, where a is given by equation (5).
Hence, if a stock out does not occur, then, using (3), (5) and (7), the utility of household i
in period t, conditional of the purchase of Qijt, can be written as:
(9) 221 ]
2[]
2)(
[1 it
ijtitijtit
ijtijtitijtijit
itRQIc
QIacQPY
aQI
U ���
�
���
�
�
�
� �2
321 ijtijtit QQD ��� ���
In this case, the inventory of household i in the following period t+1 will be
(10) itijtitit RQII ����1
and the quality-weighted inventory will be
(11) ]11][1[1 1 aQII ijtijtitit ���
�
�
However, if the total amount of the storable good, Iit+ Qijt, is less than the minimum usage
requirement Rit, all the inventories are used and a stock out occurs. In this case the utility of
household i in period t can be written, using (3), (5), (6) and (7), as:
(12) 221 ]
2)(
[]2
)([1 ijtitijtit
ijtijtitijtijitit
QIac
QIacQPYQIU
�
�
�
����� �
� � )]]([[ 102
321 ijtititijtijtit QIRssQQD ������� ���
Due to the stock out, both Iit+1 and I1it+1 are equal to zero.
16 It is worth emphasizing that the dependence of the household’s state on the quality weighted inventory I1 is one of the critical ways in which our model differs from that of Hendel and Nevo (2001b). In their model, a household gets the same utility from consuming any brand, so households do not care about the quality of their inventories, only the level.
16
2.3.3. Identification
At this point, we have laid out all the equations of our structural model of household
behavior. A formal analysis of identification is not feasible for a highly complex non-linear
model like ours. However, in Appendix A we present an intuitive discussion of how the key
model parameters are pinned down by patterns in the data. To summarize, note that the key
structural parameters are the preference weights, �, the means of the log usage requirements, �,
the inventory carrying cost parameters, c, the fixed cost of purchase parameters, �, and the stock
out cost parameters, s. The discussion in Appendix A includes simulations that show how
changing each of these parameters leads to different types of effects on household behavior,
suggesting that each parameter is separately identified. An exception is the linear term in
inventory carrying costs, c1. As we describe in Appendix A, this has almost identical effects on
behavior as the linear term in the fixed cost of a purchase, �1. Thus, we fixed c1=0.
2.4. The Price Process
A key component of our model is the vector stochastic process for the prices of each brand/size
combination. In order to have confidence in our model’s predictions of how price expectations
affect brand and quantity choice dynamics, it is important that our assumed price process be
realistic.17 Thus, our price process must capture three important features that are typical of
observed price data for most frequently purchased consumer goods: 1) prices typically are
constant for several weeks, followed by jumps, 2) the probability and direction of jumps depends
on competitor prices, and 3) the direction of jumps depends on own lagged price. To capture
these features of the data we specify a multivariate jump process that appears to be somewhat
novel in its structure.
A key problem that we face is that the number of brand/size combinations is very large
for the typical frequently purchased consumer good (e.g., in the case of ketchup it is 16). And per
ounce prices for the same brand typically differ across sizes. This creates two problems. First, it
is not feasible to estimate a vector price process including each of the 16 brand/size
combinations.18 Second, if the price process exhibits persistence, so that current prices alter
expected future prices, the expected value of the household’s next period state will depend on the
17 Since we assume that households use this process to forecast future prices, it is important that this price process fit the data well. 18 One problem is that a substantial proliferation of parameters would be entailed (i.e., consider the size of the variance/covariance matrix of the vector of price innovations).
17
current price of each brand/size combination. Thus, we must keep track of an infeasibly large
number of state variables when solving the household’s dynamic optimization problem.
To arrive at a practical solution of this problem, we exploit a common feature of most
frequently purchased consumer goods categories. In most categories, there is one clearly
dominant (or most popular) container size. That is, the large majority of sales are for a particular
size. Thus, our solution is as follows: First, we estimate a vector process for the prices of the
most common size (e.g., 32 ounces in the case of ketchup) of the alternative brands. This process
captures the patterns of persistence and competitor reaction observed in the data. Second, we
specify (for each brand) a process for the differentials of the per ounce prices of the “atypical”
sizes relative to the most common size. We assume that the price differentials between the
atypical sizes and the most common size are iid over time (except for constant mean differentials
that capture the fact that per ounce prices differ systematically across sizes).
The assumption that price differentials between the atypical sizes and the common size of
each brand are iid over time greatly simplifies the solution of the dynamic optimization problem.
It means that the only state variables we need to keep track of are the prices of the common size
of each brand. Without this assumption, the estimation of our model would be completely
infeasible.19
To proceed, we first specify the price process for the most common size of each brand,
and then specify how price for atypical sizes move relative to the common size prices. The price
of the most common size of brand j, denoted by c, is assumed to stay constant from one week to
the next with probability � . That is: jt1
)()( 1, cPcP tjjt �
� with probability � , for j=1,J. jt1
Thus, the probability of a price change is � . In this case, the process is posited to be jtjt 12 1 ���
(13) ��
��
����
4
11,21,10 )]}(ln[)4/1{()](ln[)](ln[
ljttltjjjjt cPcPcP ����
where
(14) ])()(exp[1
])()(exp[2
12110
212110
1��
��
�����
����
�
tjttjtjj
tjttjtjjjt PPPP
PPPP���
���
� ; ��
�
� �
4
11,1 )4/1(
jtjt PP
19 In our view, the assumption is probably fairly innocuous. Since most purchases are of the most common size, value functions should not be too sensitive to prices of atypical sizes.
18
and the vector of price shocks has a multivariate normal distribution
),0(~ �Nt� .
Note that equation (14) specifies the probability of a price change as a logistic function.
To capture competitive reaction, the probability that a brand changes its price is allowed to
depend on the difference between the brand’s current price and the mean price of the other
brands. Equation (13) specifies that if prices do change they follow an autoregressive process (in
logs). Competitor reaction is captured by entering the mean price of other brands in the process.
Finally, the price process for the atypical sizes is specified as:
� � � � � � � �zvcPzbzbzP jtjtjjjt ��� ln)(ln 21 ,
where c again indicates the common size and z indexes the atypical sizes. We also assume
� � � �2,0~ vjt Nzv �
The price process parameters are estimated in a first stage using the price data, prior to
estimation of the choice model. The price process parameters are treated as known in the second
stage, at which point we plug them into the consumer’s dynamic optimization problem. The
vector autoregressive jump (or switching) process for prices of the common size is estimated by
maximum likelihood, while the price processes for the atypical sizes are estimated by OLS
regression.20
Our model makes the strong assumption that consumers observe the price process
realizations each week. We considered two types of alternatives to this basic model. One is a
model in which consumers only see prices and can only make purchases in the weeks in which
they visit a store. Then the dynamic optimization problem can be simply modified by specifying
a weekly probability of a store visit. An agent at time t who is in a store and observing a set of
prices must take into account probability he/she might not visit a store next week (and therefore
won’t be able to make a purchase or see prices next week) when deciding whether to purchase at
time t. But we found that this model produced essentially identical results to our model, because
the large majority of consumer’s visit a store in the large majority of weeks.
20 We estimate the price process faced by households, which is subtly different from the price process that exists in stores. That is, we view the sequence of stores visited by the households each week as a process that is exogenous to the brand and quantity choice process. This random variation in the store visited then generates additional price fluctuations, which are subsumed in the price process we estimate. This assumption of exogeneity of the store process would probably not be a good assumption for big ticket items (say meat) where price advertising might influence the store one visits. We doubt that this is an important factor for inexpensive items like ketchup.
19
A second more extreme alternative is to assume that consumers only see prices in the
weeks they actually purchase the good. This could be rationalized by a model in which
consumers first decide whether to buy the good in a given period, and only then go to the store
and observe prices. But we reject this option out of hand, because such a model could not
possibly explain the purchase acceleration effects that are clearly present in the data.
Having completely described our model, we can provide some discussion of how it
differs from that of Hendel and Nevo (2001b). Their model is in many ways similar to ours, but
they make two key simplifying assumptions: 1) that the utility from a brand is derived entirely at
the moment of purchase, and 2) that the state of the price process can be described by an overall
price index for the category. Hence, a household’s state depends only on its total inventory (and
not how it is allocated among different brands), and the category price index. The first
assumption allows Hendel and Nevo to achieve a separation of the brand choice and quantity
choice problems - households solve a dynamic optimization problem to choose optimal quantity
each period, and then choose brands (conditional on quantity) in a static framework.21 Note that,
given the second assumption, the number of state variables is reduced to just two – the total
inventory and the category price index.
While the Hendel-Nevo approach leads to an important computational simplification, this
of course comes at some cost. First, the assumption that the vector price process for all brands
can be summarized by a single price index will fail if the stochastic process for price differs
across brands (e.g., if different brands have different degrees of price persistence). This is
particularly problematic if one wants to consider policy experiments in which one brand alters its
pricing policy. Second, the complete separation of the brand and quantity choice problems
breaks down if there is unobserved heterogeneity. In that case, the distribution of brand
preferences in the selected sample of consumers who chose to buy a positive quantity in any
21 Taken literally, Hendel and Nevo’s first assumption implies that brands are identical in attribute space (so they all generate the same utility when consumed), but that households’ perceptions of brands alter which brands they like to purchase. Such perceptions might be generated by “persuasive” or “image” advertising. This contrasts with the familiar “characteristics approach,” in which the utility of a brand is derived from its consumption. Brands occupy different positions in an attribute space (e.g., taste, texture, etc.) and a consumer’s brand preferences are determined by his/her tastes for attributes (see Lancaster (1966), and, for examples, see Elrod and Keane (1995), Berry, Levinsohn and Pakes (1995) and Erdem and Keane (1996)). However, if discount rates are small, the brand intercepts estimated by Hendel and Nevo will approximate the expected present value of the utility flow from consuming the brand. So in many contexts this distinction may be unimportant.
20
given period will, in general, differ from population distribution of brand preferences (in a way
that depends on prices).22, 23
In contrast, we model price processes at the brand level, and we assume that consumers
get utility from consumption (rather than purchase) of brands. This makes our computational
problem substantially more difficult than that faced by Hendel and Nevo, since prices of all
brands and the entire composition of inventories are state variables. The main advantages of our
approach are: 1) we can allow for more flexible models of the price process, and 2) we have
more flexibility in allowing for unobserved heterogeneity in brand preferences.
Unobserved heterogeneity in brand preferences can have important implications for how
consumers optimize in the presence of inter-temporal price variation. To give just one example,
consider a consumer who is very loyal to a particular name brand. Suppose he/she is low on
inventory, and faces a situation where current prices are high for his/her preferred name brand.
This consumer has an incentive to buy a small quantity of the inexpensive store brand in order to
tide him/herself over until a future time when the price of his/her favorite brand is lower,
anticipating that he/she can “stock up” on the favorite brand at that time.24 Such behavior
depends crucially on unobserved heterogeneity that generates a strong preference for a particular
name brand.
2.4. The Household’s Dynamic Programming Problem
The household’s optimal purchase timing, brand choice and quantity decisions can be
described by the solution to a dynamic programming problem (see, e.g., Rust (1987), Pakes
(1987), Wolpin (1987), Eckstein and Wolpin (1989), Erdem and Keane (1996)) with inventory
Iit+1, quality weighted inventory I1it+1 and prices of the common size, Ptj for j=1,J, as the state
variables.25 We assume that households solve a stationary problem.26
22 Of course, Hendel and Nevo can generalize their price process to include brand specific prices as state variables. They can also allow for heterogeneity by estimating the brand choice model with fixed effects and then solving person-specific DP problems (conditional on a consumer’s fixed effects). But with these complications the computational simplicity of their approach is lost. 23 As we noted in section 1, this is a source of bias in any estimation of price elasticities of demand based on static choice models. 24 In the data we examine, the store brand is indeed bought in small quantities much more commonly than the name brands. This is precisely the mechanism our model uses to explain this phenomenon. 25 In describing the households’ problem, we suppress the dependence of the value functions on household type, which depends on preference type and usage rate type. We also suppress the dependence of price on the household i that arises because different households shop in different stores.
21
Households are assumed to make their purchase decisions after they observe the prices at
period t but before they observe their period t usage requirement (Rit).27 Now let us define the
value function associated with the purchase of brand j and quantity Q before the realization of
the usage requirement to be
),(),,1,(),1,( 1 QjeRPIIVEPIIV itittititjQtRtititjQt t �
��
where eit is a stochastic term known to the household at the time of purchase but not observed by
the analyst. To obtain multinomial choice probabilities (see McFadden (1974), Rust (1987)), this
error term will be assumed extreme value and i.i.d. distributed. � denotes the inverse of the scale
factor (see Swait and Louviere (1993)), which is proportional to the standard deviation of the
extreme value distributed error term.
The value function associated with the above problem for household i at period t is
(15) )},(),,1,({),1,( 1, QjeRPIIVEMaxEPIIV itittititjQtRQjetitit tit �
��
In writing the alternative specific value functions VjQt(Iit, I1it, Pt, Rit) there are two cases
to consider. First, if Iit+ Qijt >Rit there is no stock out. In that case, using (5), (8) and (9) and
applying Bellman's principle, the value function associated with brand j for household i at time
period t is
221)()1(),,1,()16( ititijtjtit
ijtit
itijtijitittititjQt IcIcQPY
QIR
QIRPIIV ����
�
�� �
� � ),1,( 1112
321 1 ����
���� tititPijtijtit PIIVEQQDt
����
with Ii,t+1 given by (10) and I1i,t+1 given by (11).
If Iit+ Qijt < Rit there is a stock out. Then, using (8) and (12), the value function is:
2211),,1,()17( ititijtjtitijtijitittititjQt IcIcQPYQIRPIIV ������ �
)]]([[)( 102
321 ijtititijtijtit QIRssQQD ������� ���
),1,( 1111 ����
� tititP PIIVEt
�
26 As described in Appendix B, we obtain a stationary solution for the value functions by artificially assuming a terminal period where all value functions equal zero, and then backsolving from that period until the state specific value functions converge to a fixed point. 27 As described in section 2.2.1, the usage rate is stochastic for two distinct reasons. Conditional on the household’s usage rate type, there is an iid stochastic shock to the usage rate each period. But also, the household’s usage rate type varies over time according to a Markov process. We subsume both types of uncertainty when we take the expectation over the usage rate realization.
22
and next period inventory levels will be such that Iit+1=0 and I1it+1=0. Note that V0t, the value
of the no purchase option, is obtained just by substituting Qijt = 0 in either equation (16) or (17).
Equations (16) and (17) capture the notion that households may not make the choice that
maximizes the expected time t payoff, but rather will also consider the consequences of their
time t decisions for expected future payoffs. For example, if a household expects that a
substantial price cut for their favorite brand is likely at t+1, it may be optimal to make no
purchase at t, even if this means running a high risk of a stock out, because it is optimal to try to
arrive at t+1 with inventories as low as possible. On the other hand, if a substantial price cut for
a favorite brand occurs at t, it may be optimal to buy heavily - thus incurring substantial carrying
costs at t and in the near future - due to the expected utility flow from consuming the brand over
the next several periods.
We are now in a position to write out the probability that a household chooses to buy a
particular brand/size combination conditional on its state (which includes inventories and the
current price vector). Denote by dijt an indicator equal to 1 if household i buys brand j at time t,
and equal to 0 otherwise, and let di0t=1 denote the no purchase option. Since we have assumed
that the alternative specific taste shocks �ijt in equation (15) follow an i.i.d. extreme value
distribution, the probability that household i purchases brand j in quantity Q at time t is given by
a multinomial logit type expression:
(18) � �� �� �
� �� �� ��
�
��
QJlRPIIVE
RPIIVEPIIQdob
ittititlQtR
ittititjQtRtititijtijt
it
it
;..,,0,,1,exp
,,1,exp],1,|),1[(Pr
With regard to the summation in the denominator, Q must belong to a discrete set of available
sizes, which may in general be different for every brand j. Also note that l=0 corresponds to the
no purchase option, and there is slight ambiguity in notation because in that case V does not have
a Q subscript. Finally, note that the probability of no purchase is obtained by substituting V0t for
VjQt in the numerator of (18).
2.5. The Solution of the Dynamic Programming Problem
Given the very large number of points in the state space, we do not solve for the value
function at each point. Instead, following Keane and Wolpin (1994), we evaluate the value
function only at a finite grid of points, assigned randomly over (I, I1, P) space. We then fit
23
polynomials in (I, I1, P) to the values on these grid points, and use them to interpolate the value
function at points outside the grid points.
Using a polynomial in state variables to approximate the value function has an additional
advantage: the integrations of the value function with respect to price shocks that appears in (16)
and (17) can be done separately for each polynomial term in price that appears in the
approximation. These integrations can be done analytically, since the price shocks in (13) are
normal. Also, the integration with respect to the usage requirement shocks that appears in (15)
can be done simply using quadrature integration.
We describe the details of the solution of the dynamic programming problem and of our
approximation methods in Appendix B.
2.6. The Likelihood Function and the Initial Conditions Problem
In our model, household choices are stochastic from the perspective of the
econometrician for four reasons: The econometrician does not observe a household's preference
type, its usage rate type, or its inventory levels. And furthermore, the econometrician does not
observe the idiosyncratic extreme value distributed taste shocks for brand-size combinations.
The choice probabilities given in equation (18) assume that only the taste shocks are not
observed by the econometrician. However, we need to form choice probabilities by integrating
over all the state variables that are unknown to the econometrician. Thus, we also need to
integrate over the latent taste types, usage rate types and inventory levels.
Note that we face an initial conditions problem since we do not know the inventory levels
of households at the start of the data set (see Heckman (1981)). We integrate out the initial
conditions in the following way: We assume that the process had a true start that occurred t0
periods prior to the start of our data, so that households had zero inventories at that point. Call
this t=1. Our model specifies probabilities that each household is each usage rate type at t=1
(these were denoted �1 through �4 in section 2.3.1). Conditional on an initial usage rate type and
a preference type, we simulate the household’s purchase and consumption process for t0 weeks
(this requires us to draw prices, usage rates and usage rate types), bringing us up to the start of
the observed data.28 Call the first period of observed data t=t0+1. Doing this M times, we obtain
28 To draw prices we use a block bootstrap in which we sample 10 week long sequences of prices from the actual price data. This was done in an attempt to retain the serial correlation properties of prices present in the data.
24
M simulated initial inventory levels and initial usage rate types.29 This process is repeated for
each of the L possible initial usage rate and preference types, and for each household in the data.
Thus, for each household we get L�M draws of initial inventories.
Suppose that we observed consumption of households during the sample period, which
runs from t=t0+1 to t=T. Then we could form the simulated likelihood30 of household i’s
observed choice history as follows:
� ���
�
�
�
��
�
�
�
�
�
�
�� � � ��� � �
�
��
K
k
L
l
M
m
Tt
ttkitt
mlkit
mlkitit
mlkititijtijtlki PIIIIIQdob
Mmlil
1 1 1 1
0000
00000
,,),1,(1),(,Pr1log ��L
where denotes the observed choice for household i at time t, denotes the observed
quantity for household i at time t, and denotes the price vector faced by household i at time t.
is the population proportion of taste type k, � is the probability the initial usage rate type is l,
and is the vector of taste parameters for taste type k. Here, denotes the inventory
level of the household i at time t, conditional on simulation m of the initial inventory level and
type, as well as on the households choice and consumption history up to time t. The object
is similarly defined. l is the usage rate type for the household at t
0ijtd
k
,0
I
0ijtQ
)0
mlkit
0itP
mlit0
k�
1itI
l
�
( mlkitI
(it II
)10
mlkit 0, according
to simulation m and conditional on draw l for the initial usage rate type.31
Unfortunately, we also face a problem of unobserved endogenous state variables, because
we do not observe households’ usage rate realizations (or equivalently, their consumption levels)
even during the sample period. Thus, even if we knew the initial inventory level at the start of the
observed data, we could not construct in-sample inventory levels. We deal with this problem by
simulating each of the M inventory histories constructed above forward from t=t0+1 to t=T.
Then we form the simulated likelihood contribution for household i’s observed choice history as
follows:
� ���
�
�
�
��
�
�
�
�
�
�
�� � � ��� � �
�
��
K
k
L
l
M
m
Tt
ttkit
mlkit
mlkitijtijtlki PIIQdob
Mmlitl
1 1 1 1
0000
0
,,,1,,Pr1log ��L
29 In our application, we set M=10 and t0 = 246 (which is equal to twice the number of weeks of observed data). Also note that L=4�4=16. 30 See Keane (1993, 1994) for a discussion of simulated maximum likelihood methods for discrete panel data. 31 In forming the likelihood, we have left the integration over the latent usage rate types from time t0+1 through t0+T implicit.
25
where is the inventory level at time t for household i of taste type k and initial usage rate type
l, according to the mth draw sequence. The quality weighted inventory I is defined similarly.
And
mlkitI
mli
mlkit1
tl denotes household i’s usage rate type at time t according to draw m and conditional on
initial type l.
3. Data Description
We estimate the model introduced in Section 2 on A.C. Nielsen scanner panel data from
Sioux Falls, SD.32 The data set contains 2797 households and covers a 123 week period from
mid-1986 to mid-1988. Every market of any significant size in the city of Sioux Falls was
included in the study, so that we should have fairly complete data on the purchases of the
participating households.33
Three national brands (Heinz, Hunt’s and Del Monte), together with Store brands,
capture more than 96% of total sales in this market. We therefore restricted the analysis to these
four brands, and eliminated households that bought other, minor, brands. Among these four,
Heinz is clearly the dominant brand, with roughly a 66% share of all purchases, followed by
Hunts at 16%, Del Monte at 12% and Store brands at 5%. The sizes available are 14, 28, 32, 40
and 64 ounces. But Hunt’s is not available in 14 and 28 ounce sizes, and the Store Brands are not
available in the 40 and 64 ounce sizes. Of all the 16 available brand/size combinations, Heinz 32-
ounce is the market share leader with 36% share.
We wanted to limit the sample to households who are regular ketchup users because it
seems unlikely that our model would be relevant for households who are not regularly in the
market. A careful inspection of the data revealed that some households would be heavy ketchup
users for several months, and then seem to never purchase again. We are uncertain if this is
because these households actually stopped buying ketchup, or perhaps because of some problem
with the data.34 In order to obtain a sample of households who appeared to be regular ketchup
buyers throughout the 123 week period, we subdivided the period into three 41 week sub-
periods. Then, we took only households who bought at least once during each sub-period. This
reduced the sample size from 2797 households to 996 households.
32 Data is also available for Springfield, MO for the same sample period. 33 We will of course miss purchases that were made out of town. 34 We speculate that some households may have moved out of Sioux Falls, but that this wasn’t recorded, or perhaps that the ID cards malfunctioned for some households.
26
Figure 1 reports the distribution of households by total number of ketchup purchases
during the 123 week period. We discovered that with only 4 usage rate types our model had
difficulty simultaneously fitting the fat right tail of very heavy ketchup users, along with the
large number of light users. This problem is compounded by the fact that, as we noted earlier,
households usage intensity often seems to vary greatly over the 123 week period. Thus, our
usage rate heterogeneity distribution has to play the dual role of explaining the dispersion in
purchase frequency across households (Figure 1), and the heterogeneity within households in
purchase intensity over time. Adding more usage rate types would solve the problem, but
computational barriers precluded us from pursuing that course.
Hence, we decided to further screen the sample down to households who bought at least 4
times and bought no more than 16 times over the 123 week period. This further reduced the
sample size from 996 to 838.
We also had to decide on which purchase quantities would be included in the choice set.
As we noted, there are 5 sizes of ketchup container (14, 28, 32, 40 and 64 oz.), but households
could purchase other quantities by buying multiple containers. However, we found that 7 options
accounted for more than 99% of all ketchup purchases: 1) buy a single container of one of the 5
sizes, 2) buy two bottles of the 14-ounce size, or 3) buy two bottles of the 32-ounce size. Since
option (2) generates a 28 oz purchase, and option (3) generates a 64 oz purchase, we decided to
limit the discrete set of quantities that any household can buy to just {14, 28, 32, 40, 64}.
A feature of the data is that not every brand size/combination is available in every store in
every week. Table 1 reports the sample frequencies with which each brand/size was present in
the choice sets of the households in the data (conditional on the stores they visited each week).
This variability in the choice sets was accounted for in both the solution of the DP problem and
the construction of the likelihood for our model. We ignored this in the presentation of the
model, because it would be notationally cumbersome. Essentially, we assume the households in
our model know the probabilities in Table 1, and that they take these into account when
constructing their expected value functions.
Tables 2 and 3 contain some descriptive statistics about prices. Table 2 reports the mean
(offer) price of each of the 32 oz sizes in cents.35 Note that Heinz, the most popular brand, is also
35 The price variable used in the estimation is the price paid before coupons (i.e., the shelf price). Including the redeemed coupon value in the price of the purchased brand would create a serious endogeneity problem. This is
27
the most expensive. Table 3 reports the mean price per oz differentials between the various sizes
and the 32oz size. Notice that, in most cases, the 32 oz size is actually cheaper, on a price per oz
basis, than the larger sizes.
4. Empirical Results
4.1. Parameter Estimates for the Price Process
Table 4 reports the maximum likelihood estimates of the parameters of the price process
for the per ounce price of the 32 ounce sizes. The top panel of the table reports the parameters in
the logit for the probability that price remains constant from one week to the next. The most
interesting coefficient here is �2, the coefficient on the squared difference between own price and
mean competitor price. This term is negative, indicating that when the price differential is large
the contribution of this term to the logit becomes a large negative. Thus, a brand’s price is less
likely to stay constant if it departs greatly from competitors’ prices.
The bottom panel of Table 4 reports the parameters of the autoregressive process for log
per ounce prices in the event that there is a price change. Note the autoregressive coefficient on
own lagged price is .4473, while the coefficient on the average price of competitors is .1482.
This is again consistent with competitor reaction, since it implies that Pt+1 tends to be higher
relative to Pt if competitors’ prices are higher.
Also interesting are the covariances between the price shocks. Note that the covariances
among the price shocks for the three national brands (�23, �23, and �23), are very small, and in
two out of three cases negative. This suggests that when brands change prices simultaneously in
a given week, there is no clear tendency for the prices to move in the same direction. This
suggests that common demand shocks are not driving the price changes, which is consistent with
our argument that price movements are largely exogenous from the point of view of consumers.
Table 5 reports the OLS estimates of the processes for how the per ounce prices of the
“atypical” sizes differ from that of the common 32 ounce size. Interestingly, the fact that the
slope coefficients are in many instances small suggests that these prices do not move very
because we do not observe what coupons the households could have used for the brands they chose not to buy. Including the coupon value only in the price of the brand actually bought is like including a dummy for the brand purchased (interacted with coupon value) as an explanatory variable in the choice model. That is, one is including a transformed version of the dependent variable as an independent variable!! While this has often been done in scanner data research, it is clearly a serious misspecification. Erdem, Keane and Sun (1999) show that it leads to serious exaggeration of price elasticities of demand.
28
closely together.36 This again suggests that brand specific demand shocks are not what drives
price fluctuations.
4.2. Parameter Estimates of the Choice Model
Table 6 presents the simulated maximum likelihood estimates of our dynamic model of
consumer choice behavior. Consider first the taste parameter estimates for the four taste types.
These are interpretable as cents per ounce. Thus, type 1 households receive a monetary
equivalent utility of 4.09 � 32 = $1.31 from consuming a 32 oz container of Heinz. Type 1’s have
a clear preference for Heinz over the other three brands. And type 1’s account for 51% of the
population, which is consistent with Heinz’ dominant position. This “loyal” type will buy Heinz
almost exclusively.
Type 2’s and 3’s also prefer Heinz to the other brands, which illustrates just how
dominate Heinz is in the this market. However, type 2’s like Hunts almost as much as they like
Heinz, and type 3’s like Del Monte almost as much as they like Heinz. Type 2’s will tend to
switch over time between Heinz and Hunts, while type 3’s will tend to switch between Heinz and
Del Monte.37 The type 4’s, who make up only about 4% of the population, like the store brand
much more than do the other types. They will tend to switch between Heinz and the Store brand
over time.
The next section of Table 6 contains the estimates of the parameters that characterize
stock out costs, inventory carrying costs and the fixed cost of a purchase. Note that the linear
inventory carrying cost term was pegged at zero for identification reasons, as discussed in
Appendix A. The quadratic inventory term (.001157) implies that the cost of carrying a stock of
32 ounces is only about 1 cent per week. Based on this, inventory carrying costs may seem to be
trivial. It should be noted, however, that the quadratic term would become important if
households tried to hold very large inventories (e.g., at 100 ounces it becomes 11.6 cents per
week, and at 200 ounces it becomes 46 cents per week). Thus, this parameter plays a key role in
the model (i.e., simulations of our model imply that households rarely hold inventories in excess
of 64 ounces, and practically never hold more than 80 ounces).
36 This is a weakness of our approach, since it means our model of the price process fits the behavior of the “atypical” sizes less well than we would like. But, as noted in Section 2.4, we expect that value functions will not be too sensitive to the price process for the atypical sizes since they are bought much less frequently than the 32oz size. 37 Elrod and Keane (1995) and Keane (1997a,b) discuss how brand switching patterns tend to identify distributions of consumer taste heterogeneity.
29
The stock out cost is about 12 cents. In contrast, the fixed cost of making a purchase of a
32 ounce size is 228 – 4.73 (32) + 0.06 (32)2 = $1.38.38 This slightly exceeds the typical price of
the 32oz size. This estimate seems quite reasonable if one interprets the fixed cost as consisting
primarily of the utility cost (i.e., time cost) of going to the store to make the purchase. Given the
low stock out cost and the large fixed cost of making a purchase, it is not surprising that
simulations of the model imply stock outs are extremely common (see Section 4.3 below).
We turn next to a discussion of the usage rate parameters for each of the four usage rate
types. Type 1’s have a very high usage rate – about 23 ounces per week on average.39 But the
probability a household remains a type 1 from one week to the next is only .35. The model uses
the type 1’s to capture instances in the data where households are observed to buy large amounts
of ketchup in consecutive (or nearby) weeks. We speculate that these unusual episodes are
probably due to events like container breakage or instances where families throw large parties or
cook outs.
Type 2’s and 3’s exhibit much more moderate usage rates, and also much greater
persistence over time. For instance, type 2’s use about 8 ½ ounces per week on average. They
have a week-to-week probability of staying type 2’s of .9958, which implies there is about a 20%
chance they change type within a year. Type 3’s use about 2 ounces per week on average.
Note that usage rate parameters for type 4’s are not reported in the table. In the estimation
process, the model wanted to generate one type with a very low usage rate. This enables the
model to explain instances where households go several months without buying any ketchup.
Thus, at some point in the estimation process we simply fixed the usage rate for type 4’s at zero.
The last set of estimates, reported at the bottom of Table 6, are the probabilities that a
household is each of the four usage rate types in the initial week of the data. The most common
initial type is actually the zero usage rate type (e.g., 34.2% if family size is set to zero). As we
would expect, the family size coefficient suggests that larger families are more likely to be the
higher usage rate types (initially). The estimate implies that the probability of being a zero usage
38 Note that the estimates of the quadratic in container size implies that the fixed cost of purchasing the 32 oz size is lower than the fixed cost of purchasing any other size. This seems plausible, given that the 32 oz size is typically prominently displayed in the store (sometimes including end of aisle displays, in aisle displays, etc.), while other sizes may take more effort to locate. Obviously, our fixed cost parameters are capturing time and search costs, not the physical effort involved in carry containers (which is presumably relatively trivial). 39 To obtain this figure, use the �1 and �1 from Table 6, and plug them into the formula exp(�1 + �1
2/2).
30
rate type drops by about 3.5% with each additional family member (e.g., the probability of being
the zero usage rate type drops to 29.8% if family size is 4).
4.3. Goodness of Fit
Table 7 compares the sample choice frequencies and simulated choice frequencies for all
brand/size combinations. Overall, the fit of the model seems remarkably good on this dimension.
The probability that a household makes a ketchup purchase in any given week in the data is
6.768%. Simulation of our model generates a probability of 6.771%. The model also fits the
brand shares extremely well. For example, the Heinz share is 66.4% and the model predicts
64.6%.
The only dimension in which the model (slightly) fails is generating the size distribution
of purchases. The 32 ounce share is slightly underestimated (61% in the simulation vs. 64% in
the data) and the 28 ounce share is also slightly underestimated (13.7% in the simulation vs.
17.0% in the data). Both these errors get pushed into the 40 ounce share, which is seriously
overestimated (11.4% in the simulation vs. 5.4% in the data).
Obviously, inventories are unobserved, so we cannot compare the model’s predictions to
the data. The simulation implies that households carry a mean inventory of 7.5 ounces, and that
they are stocked out (have no ketchup at all) in 2/3 of the weeks. At this point it is useful to recall
that, in our model, households adjust consumption rates along the extensive margin (i.e.,
percentage of weeks they have ketchup available to consume) rather than along the intensive
margin (i.e., rate of ketchup consumption in weeks when it is available). Thus, price changes
effect consumption via their effect on stock out frequency.
An interesting aspect of the data is that the share of the 14 ounce size in total brand sales
is much greater for the Store brand (29%) than for the name brands (7% for Heinz and 4% for
Del Monte). The model captures this pattern quite well, despite the fact that there is no specific
parameter that could pick it up (i.e., we do not have brand/size specific taste parameters). Thus,
the pattern is generated by the basic structure of our behavioral model itself. We can show in a
greatly simplified version of our model that if a household has low inventory (i.e., it is at risk of
stocking out) and the prices of preferred name brands are high, it is optimal to buy a small
amount of the cheapest brand as a stop gap measure while waiting for prices to fall. This basic
mechanism presumably carries over to the more complex model estimated here.
31
Figures 2, 3 and 4 provide evidence on how the model fits choice dynamics.40 Figure 2
compares the simulated and actual distributions of inter-purchase times (in weeks). The main
failure of the model is that it somewhat underestimates the frequency of very short inter-purchase
spells. For instance, the percent of the time that people buy again in just one week is 3.8% in the
data vs. 2.7% in the simulation. By four weeks the model predicts 5.3% vs. 5.9% in the data. But
other than that, the agreement of the simulated and actual distributions is quite impressive. The
modal inter-purchase time is 6 weeks, and this is correctly predicted by the model. The model is
also accurate regarding the amount of mass in the vicinity of the mode. In data, 18.8% of spells
are in the 5 to 7 week range, compared to 17.7% in the simulation. The model (very) slightly
overestimates the percent of inter-purchase spells in the 8-22 week range, and is quite accurate
for spells of over 22 weeks.
Another way to look at the data is to look at the survivor function for no-purchase spells,
which is reported in Figure 3. Here, the agreement between the model and the data looks very
impressive. Consistent with the observations made above, the simulated survivor function from
the model is slightly above that in the data in weeks 1-16, because the model predicts too few
short spells. And the simulated survivor function drops a bit below the data in the 21-37 week
range – because too many spells are predicted to end in that range. But the divergence between
the data and simulated survivor functions is never more than a few percent.41 In the data the
survivor function first drops below 50% at 10 weeks (i.e., 47.8% of no-purchase spells survive
more than 10 weeks). The model survivor function implies that 50.2% of spells survive past 10
weeks, and dips below 50% at 11 weeks. The model and data survivor functions both drop below
20% at 18 weeks.
Figure 4 reports hazard rates for the hazard of making a purchase. Again the empirical
and simulated hazard rates line up quite well. The hazard rate for the data is rather jagged due to 40 Of course, in the data, some spells are left or right censored. To make the simulations of the distribution of interpurchase times, the survivor function and the hazard comparable to those in the data,we imposed the same censoring on the simulated data. However, we found that this led to only trivial changes in the simulated distributions. This contrasts with the usual experience with unemployment duration data, where truncation typically has large effects. The reason for the difference lies in the different nature of these two types of data. In unemployment spell data, the sample usually consists of people who became unemployed in a particular week. Thus, the finite length of the sampling period leads to right censoring of longer spells. In our data, in contrast, the sample begins at random points during no-purchase spells of households. And the sampling frame of nearly three years is long relative to even the longest no-purchase spells. This means that short spells are just as likely to be right censored as long spells when the data set ends. 41 The maximum divergence is at week 7. In the data 61.9% of no-purchase spells survive past week 7, and the model predicts 65.8%.
32
noise, especially after about 30 weeks, since less than 10% of all no purchase spells survive that
long (see the survivor function). The model predicts that purchase hazard is quite low
immediately after a purchase, and then rises to the vicinity of 8% after about 7 weeks. It then
stays fairly flat at that level regardless of spell length. Note that empirical hazard is very similar
to the simulated hazard up through week 16, and by that point over 70% of spells are ended (see
survivor function). The empirical and simulated hazards diverge a bit after week 16. The
difference is that, while the simulated hazard stays near 8%, in the actual data the hazard sags to
the 6-7% range in weeks 16-32, and after week 35 it averages around 10%.
Table 8 reports on how the model fits the distribution of accepted (per ounce) prices. The
top two panels of the table contain mean offer prices from the data vs. simulation of the model.
These are virtually identical. The bottom two panels contain mean accepted prices from the data
vs. the simulation. The mean price for each brand/size combination is reported as a price per
ounce. For example, for Heinz, the mean offer price in the data is 3.596 cents per ounce for the
32 ounce size, or $1.15. The mean accepted price is $1.12. In the simulation, these figures are
$1.15 and $1.11, respectively. Note that mean accepted price is only a few cents below mean
offer price, which is consistent with the fact that a large fraction of consumers have a strong
preference for Heinz, thus isolating it from strong price competition.
As we would expect, differentials between offer and accepted prices are generally much
larger for Hunts, Del Monte and the Store Brand. For example, for Del Monte, the mean offer
price in the data is $1.05 and the mean accepted price is 96 cents. The simulation also generates
predictions of $1.05 and 96 cents, respectively. For the Hunts 32oz, the offer/accepted
differential is about 5 cents in both the data in the simulation. Overall, the fit of the model to the
accepted price distribution is remarkably good.42
Finally, Table 9 compares the brand transition matrix in the data vs. that generated by
simulation of our model. Some features of the transition matrix for the data are quite striking.
First, note that a household that buys Heinz on a given purchase occasion has a 79% probability
of buying Heinz again on the next purchase occasion. But the pattern is strikingly different for
the other three brands. For example, a household that buys Del Monte on a given purchase
occasion has only a 34% probability of buying it again on the next purchase occasion. It actually
42 The only exceptions are the Heinz 64 oz and the Del Monte 40 oz. For each of these, the model substantially under-predicts mean accepted price.
33
has a higher probability of buying Heinz (41%). Indeed, Heinz is so dominant in this market that
this basic pattern holds for all three alternative brands. In general our model fits the transition
matrix quite well, except that we understate the own transition rate for Del Monte by a third.
4.4. Policy Experiments
Our model could potentially be used to study the impact of a multitude of possible policy
experiments. In this section we report the results of two types of experiment that are of
particular interest. First, we discuss a transitory price cut experiment. This experiment is aimed
at evaluating the importance of price expectations in the determination of own and cross price
effects on demand. Second, we evaluate the effects of three types of permanent changes in
pricing policy: a permanent reduction in mean price, a permanent reduction in price variability,
and a simultaneous reduction in both the mean and variance of prices.
4.4.1. Effects of Transitory Price Changes: Evaluating the Importance of Expectations
In our first experiment we simulate the effect of a 10% temporary (i.e., one week in
duration) price cut for all sizes of the leading brand, Heinz. This change in price at time t will
alter the expected future prices of Heinz, and all the other brands. Using our model, we can
simulate the “total” effect of the temporary price cut on demand, which includes this change in
expectations. We can also calculate an “expectations fixed” effect, in which households use the
original (rather than the reduced) time t Heinz price to forecast expected future prices.
To conduct the experiment, we first generate 10,000 simulated price histories that last
246 weeks (twice the sample period in our data). We then simulate the behavior of 10,000
households, each facing one of these price processes. The simulated households are constructed
with the same type proportions as are present in the population according to our model estimates.
We start each household with zero inventories at t=1, just as when we integrate out the
initial conditions in simulating our likelihood function. We simulate the effect of a price cut at
week 80, under the assumption that the distribution of inventories would have converged to the
stationary distribution by that point. This leaves 167 weeks over which to trace out the impulse
response to the price cut. By using 10,000 simulated price histories, we essentially integrate over
the distribution of initial prices and inventories that exist at the time of the price cut, as well as
over the distribution of price changes (for Heinz and other brands) that occur after the price cut.
Table 10 reports the effects of the temporary price cut on purchase probabilities for Heinz
and all other brands in the week of the price cut, week 1, and in subsequent weeks through week
34
15. We found that after week 15 effects on demand were trivial, so we do not report them. Under
each brand heading in the table, the first column reports the “total” effect, and the second column
reports the “expectations fixed” effect. The first column indicates that purchases of Heinz
increase 41.3% percent in the week of the price cut (corresponding to an elasticity of demand of
–4). This very large own effect is consistent with a large body of work in marketing (using
scanner data) showing large effects of temporary price cuts on demand for many frequently
purchased consumer goods.
Consider now the cross-price effects. The 10% price decrease for Heinz results in
decreases in demand in week 1 of about 4% for Hunts, 3.6% for Del Monte and 3.1% for the
Store Brand, implying cross-price elasticties of demand in the range of .30 to 40.
Note that total demand in the category rises 25.3%. This indicates that the price cut for
Heinz is not just stealing customers away from the other brands. Rather most of the increase in
Heinz sales results from either “purchase acceleration” or category expansion.
Next, consider the effect of the price cut, holding expectations of future prices fixed. As
we would expect, the positive effect on Heinz sales is now greater; 45.3 % compared to 41.3 %
when expectations adjust. The reason is that, with expectations held fixed, given the high degree
of persistence in the price process, consumers expect that at t+1 the price of Heinz will very
likely be near its original (pre-promotion) level. Thus, there is an added incentive to purchase at
time t (i.e., “make hay while the sun shines”). Still, the most striking thing about this effect is
that it is rather small. The own price elasticity of demand holding expectations fixed is only 10 %
greater than the elasticity when expectations adjust.
The striking result in the table is the impact of expectations on cross-price elasticities of
demand. When expectations are held fixed, these are reduced by roughly 50 %. The expectations
fixed cross-price elasticities of demand for Hunts, Del Monte and the Store Brand are only about
.20. Two factors drive this result: 1) if Heinz’ price is lowered today it leads consumers to also
expect a lower Heinz price tomorrow. This lowers the value function associated with purchase of
any brand other than Heinz today. 2) Given the price dynamics in the ketchup market, a lower
price of Heinz today leads consumers to expect competitor reaction, so it lowers the expected
prices of the other brands tomorrow.
Table 11 reports the results of the exact same experiment, except that there we report the
effects of the price cut on quantities demanded, rather than on purchase probabilities. The basic
35
story is exactly the same. The only additional point worth noting is that the price cut for Heinz
causes both the Heinz quantity sold and the overall category quantity sold to increase by about
10% more than did the purchase incidence. This implies that, in response to the price cut,
consumers are also switching to somewhat larger quantities (conditional on purchase). This is as
we would expect, and is again consistent with purchase acceleration.
Finally, we report the effect of the temporary price cut on total quantity sold over weeks
1 through the end of the simulation (a period of 167 weeks). As a percentage of average weekly
sales, the sales of Heinz increase 35.5%, while the sales of Hunts, Del Monte and the Store
Brand decline –7.9 %, -6.9 % and –6.6 %, respectively. Overall category sales increase 20.5%.43
Thus, it is clear that the short run increase in the level of sales due to the temporary price cut is
not wiped out, even in the long run, by sales reductions in later periods. The temporary price cut
not only produces purchase acceleration, but also generates some additional Heinz and category
sales that otherwise would not have occurred.
4.4.2. Effects of Permanent Changes in Pricing Policy
The real strength of a structural approach to demand estimation is that we can forecast
how consumer behavior will respond to fundamental changes in pricing policy. In this section we
analyze three such policy changes: 1) a permanent 10% cut in the mean price of Heinz, 2) a
permanent 50% reduction in the standard deviation of Heinz prices around their mean, and 3) a
combined experiment where we lower both the mean and variance in Heinz prices. The results of
these three experiments are reported in Table 12. For both Heinz and the competitor brands we
report the percentage changes in purchase incidence, total quantity sold (i.e., sales weighted by
the container size), sales revenue, and mean accepted price.
The top panel of Table 12 reports results from a permanent 10% reduction in the mean
price of Heinz. This price reduction was applied to all sizes.44 Note that the purchase frequency
for Heinz increases 33.1%, while the total quantity of Heinz sales increases 35.6%. Thus the
price cut generates some shift towards purchase of larger sizes. As we would expect, the long run
43 Dividing these figures by 167, we see that percentage increase in Hunts sales over the whole 167 week period is less than 2 tenths of 1 percent. Since essentially all the change happens in the first several weeks, we see that asymptotically, as t grows large, the effect of the temporary price cut on the percentage increase in total sales is (of course) approaching zero. 44 To determine the new stochastic process for prices, we reduced all Heinz prices in the data by 10%, and then re-estimated the price process of Section 4.1. Since the price process includes terms that capture competitor reaction, these parameters must adjust so that the new price process generates the same distribution of competitor prices as did the original price process (despite the lower prices of Heinz).
36
elasticity of quantity demanded with respect to the permanent price cut (-3.5) is less than the
short long elasticity with respect to a transitory price cut (-4.5).
It is also interesting to compare short-run vs. long-run cross-price elasticities of demand.
Recall from Table 11 that the short-run cross-price elasticities with respect to transitory price
changes were in the range of .30 to .40. Here, we see that the long-run elasticities with respect to
permanent price changes are in the range of .75 to 1.0. Thus, the long run cross-price elasticities
are much greater than the short-run elasticities. Finally, note that the 10% price cut for Heinz
leads to a 19.7% increase in overall category demand in the long-run. As we would expect, this is
substantially less than the 31.3% short-run category expansion effect we found for a transitory
price cut in Table 11.
The finding that long-run cross price elasticities of demand greatly exceed short-run
cross-price elasticities is a key result of our analysis. This result implies that the degree of
competition between brands is substantially greater than short-run elasticity estimates would
indicate. But, in interpreting this result, it is important to bear in mind that we are not modeling
competitor reaction to the permanent change in Heinz pricing policy.45 Thus, our experiment
involves permanently lowering the price of Heinz relative to other brands. Obviously, this will
induce a certain degree of brand switching (from other brands to Heinz). In contrast, when we
simulated a transitory price cut for Heinz, this induced consumers to expect competitor reaction,
in the form of lower prices for the other brands in the future. Thus, to some extent, the transitory
price cut generates delay as opposed to switching, thus dampening the cross-effect.
There is a second mechanism that also dampens the short-run cross-price effect relative
to the long-run effect. Given a transitory price cut for Heinz, a “switch” to Heinz will, for
households with relatively large inventories, require a “purchase acceleration” ahead of the time
when they would have otherwise bought again. Such households may be deterred from switching
because it will entail extra inventory costs in the short run. With a permanent price cut this
mechanism is not operative – a household with large current inventories can simply delay the
Heinz purchase until some future point when inventories are sufficiently run down.46
45 To do so we would need to develop and estimate a market equilibrium model, which is beyond the scope of this paper. As we noted in the introduction, the technology to estimate market equilibrium models with forward-looking behavior on both the firm and consumer sides is probably several years away. 46 In other words, given a permanent Heinz price cut, a household can raise its steady state share of Heinz purchases without being subject to any short run spike in inventories. But, with a purely transitory price cut, a household whose current inventory is relatively high can only take advantage of the sale by bearing the cost of a short run spike
37
It is important to consider the implications these findings for conventional estimations of
demand elasticities using static reduced form models. The fact that elasticities with respect to
permanent and transitory price cuts differ substantially means that conventional estimates will be
quite sensitive to whether the price changes present in the data under analysis are primarily
persistent or transitory changes. Of course this is not a new point. For instance, Keane and
Wolpin (2002) recently made a similar point regarding estimates of elasticities of various
behaviors such as labor supply, fertility, marriage and welfare participation with respect to
permanent vs. transitory changes in welfare benefit rules, and, much earlier, Lucas and Rapping
(1969) considered elasticities of labor supply with respect to permanent vs. transitory wage
changes. But it appears that previous work on scanner data has not paid serious attention to this
issue.
The second panel of Table 12 reports the effects of a 50% reduction in the variance of
Heinz prices. To implement this experiment, we first calculated the mean offer price for each
separate size of Heinz container. We then compressed these offer prices around the size specific
means, and re-estimated the price process of Section 4.1. Note that the Heinz purchase
probability falls 5.3%, while total quantity of Heinz sold drops by 6.6%. This implies there is
some shift towards the purchase of smaller sizes. Heinz total sales revenue drops by 4.2%.
Revenue drops less than quantity because the mean accepted price increases by 2.6%. This is as
we would expect in a search model, given a reduction in the dispersion of the offer price
distribution.
Our final experiment was designed to determine whether it would be possible for Heinz
to increase profits by simultaneously reducing the mean and variance of prices. This corresponds
to a policy of having fewer sales, while also maintaining price at a consistently lower mean level.
This experiment is of some interest, because there was a widespread shift from a policy of
frequent sales to a policy of “every day low pricing” (EDLP) for many frequently purchased
consumer goods in the late 1980s and early 1990s (i.e., after our sample period ended).
Of course, not knowing the cost function, we can’t in general determine how changes in
pricing policy would affect profits. However, if we assume that production cost is only a
function of total quantity sold, then we can compare profits under policies that generate equal
in inventories. Intuitively, households that already have a 64 oz bottle of ketchup at home will not want to buy Heinz even if it is on sale this week, because they don’t want to waste more room in their kitchen cabinets on ketchup.
38
sales quantities simply by comparing revenues. We determined that, given a 50% reduction in
the variance of prices, Heinz would need to reduce mean offer price by 2.2% in order to hold the
quantity of sales approximately constant. This is the experiment we report in the bottom panel of
Table 12.
Note that the change in pricing policy has a positive effect on Heinz revenue, which
increases by ½ of one percent. Thus, our model of consumer demand does imply that “Heinz”
had an incentive to try this type of change in strategy. Of course, we are abstracting from the fact
that “Heinz” is not a unitary actor. Actual retail price setting for a brand involves a complex
interaction between retailers, wholesalers and manufacturers. Manufacturers use systems of
incentives to attempt to induce particular pricing strategies on the part of retailers. For
discussions of this topic, see, e.g., Lal (1990), Neslin, Powell and Schneider-Stone (1995) and
Neslin (2002). It is beyond the scope of our analysis to explain how any increase in Heinz
revenues resulting from the policy change would be distributed the various actors in the supply
chain, or to consider how such a policy change might be instigated.
Note that the Heinz pricing policy change actually reduces demand for all competing
brands and for the ketchup category as a whole. So whether a retailer would have an incentive to
try such a change in strategy is ambiguous. Of course, not knowing wholesale costs for the
various brands, we cannot determine how the policy change affects total category profits for the
retailer. Within a range of plausible estimates for markups, our estimates in Table 12 imply
reduced profits from other brands but ambiguous effects for the category as a whole.47
The reason demand for competing brands is reduced is the very dominant position of
Heinz in the market. Even the type 2 and 3 households, who account for the bulk of Hunts and
Del Monte sales, respectively, actually slightly prefer Heinz. Thus, a large fraction of their sales
derive from situations in which the Heinz price is relatively high. The variance reduction reduces
the extent of such events.48
47 For example, assuming all brands have a 20% markup, and that marginal cost is constant, the estimates imply an increase in Heinz profits of 3.2%, reduced profits from Hunts, Del Monte and the Store brand of –9%,-11% and –7% respectively, and a small decline of –0.4% in category profits. Assuming smaller markups for the smaller brands would easily swing the sign of the net category effect, as would assuming smaller markups in general. 48 Of course, the variance reduction also reduces the magnitude of Heinz sales. But there is an asymmetric effect of the variance reduction because, loosely speaking, as long as Heinz price is at or below its mean, households are much more likely to buy Heinz than the competing brands anyway.
39
Even if we adopt the abstraction of each brand as a unitary actor, our policy experiment is
also limited because it holds competitors’ pricing policy rules fixed. Thus, the policy change
could still be undesirable if Heinz expected it to induce competitor reactions that would
adversely affect Heinz profits. At best, an analysis of demand side response alone can only tell us
whether a policy change would have some potential for increasing profits given the predicted
nature of consumer reaction. It cannot reveal whether a policy change would still increase profits
once competitor reaction is factored in.
Nevertheless, our results in the bottom panel of Table 12 do suggest that a strategy based
on reducing price variance would have offered some promise. For such a strategy to be
successful, a necessary condition is that it induce a substantial increase in mean accepted price.
Notice that under the experiment, consumers are predicted to buy the same quantity of Heinz but
at a mean accepted price that is ½ of one percent higher. This may represent a significant
percentage increase in margins.
Finally, our experiment also illustrates why a price index constructed by randomly
sampling offer prices would be misleading during a period in which retailers switched to an
EDLP strategy. In our experiment the mean offer price for Heinz falls 2.2%, yet the mean
accepted price for Heinz rises 0.5%, and that for the category as a whole rises 0.85%. Thus, such
a price index would falsely imply that the price of ketchup had fallen, when in reality the
effective price of ketchup to consumers had increased.
5. Conclusion
We have shown that our dynamic model of consumer brand and quantity choice
dynamics under price uncertainty does an excellent job of fitting data on consumer purchase
behavior in the market for a particular frequently purchased consumer good, namely ketchup.
Our results indicate that most of the increase in brand sales resulting from a temporary
price cut result are due to a combination of purchase acceleration and category expansion, rather
than brand switching. Given the stochastic process for prices present in our data, cross-price
elasticities of demand with respect to temporary price cuts are modest compared to the own price
elasticities.
More generally, our work suggests that estimates of own and particularly cross-price
elasticities of demand may be very sensitive to the stochastic process for price and how
households form expectations of future prices. In particular, the magnitude of cross-price
40
elasticities of demand depends not just on the similarity of goods in attribute space, but also on
the extent to which changes in current prices affect expected future prices for the own brand and
other brands. This in turn will depend on the price process itself, which is just another way of
saying that price elasticities of demand are reduced form, and not “structural,” parameters. These
findings suggest that researchers working on merger analysis and evaluation of welfare gains
from the introduction of new goods should be careful about interpreting cross-price elasticities as
measures of the degree of competition between brands.
A second main finding of our work is that estimates of cross-price elasticities with
respect to permanent or long-run price changes are substantially greater (i.e., by a factor of two)
than estimates of cross-price elasticities with respect to transitory or short-run price changes. The
short-run estimates are dampened by the presence of both inventory carrying costs and expected
competitor reaction. For this reason, the long-run estimates provide a better measure of the
intensity of competition between brands.
Finally, we showed how a pricing policy change that involves a simultaneous change in
mean offer prices and price variability can create a substantial wedge between the change in
mean offer vs. accepted prices. Thus, price indices based on sampling of offer prices can
potentially be highly misleading as measures of changes in effective costs to consumers during
periods when price variability is changing.
41
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45
Appendix A: Identification
Our model is too complex to for us to provide analytic results on identification, so we instead provide an intuitive discussion of how the key model parameters are pinned down by patterns in the data. We also present, in Table A1, simulations of how increasing each of the key model parameters affects key features of simulated data. These simulations are useful for understanding how various parameters have different effects, and are therefore identified. First, we discuss the parameters that determine the inventory carrying cost (CC), fixed cost of purchase (FC) and stock out cost (SC). Recall that the equations for these are:
Carrying Cost: CC = c1 I + c2 I2
Fixed Cost: FC = �0 + �1 Q + �2 Q2
Stock Out Cost: SC = s0 + s1 (R – C) First, consider the linear component of fixed cost (�1) and the linear term in inventory carrying costs (c1). If the quantity Q that a consumer buys is used at a constant rate over time (i.e., R is fixed), and/or there is no discounting, it is irrelevant whether carrying costs are spread out over the period the good is consumed (reflected in c1), or whether the present value of carrying costs is born up front (reflected in �1). Thus, c1 and �1 would not be separately identified. In our model, there is discounting, and usage rates R do fluctuate over time, so the parameters c1 and �1 would have subtly different effects on behaviour, but it would not be surprising if these are hard to detect. Indeed, when we tried to iterate on both parameters, we found that the likelihood was extremely flat along a locus in (c1, �1) space, and that the two parameters would run off in opposite directions. Thus, we discovered that we cannot separately identify these two parameters, and so we constrained c1 = 0. It is comforting that this identification problem was made obvious by our search algorithm, and that such problems did not emerge for any other model parameters. Next consider �0, the constant in the fixed cost of purchase function. A large �0 would induce one to minimize the frequency of purchases, and to buy large quantities when one does buy. Thus, it is pinned down by data on the frequency and size of purchases.49 It is important to note that while a high �0 discourages frequent purchases, it does not affect or induce duration dependence in the purchase hazard. For example, with a high �0 one wants to avoid having many purchases during a year, but, conditional on the total number, one doesn’t care if purchases are spread out or close together. Indeed, simulations of our model, which we report in Table A1 indicate that an increase in �0 shifts down the purchase hazard, but has little effect on its shape. Next, consider the quadratic terms c2 and �2. These might at first appear to be subject to the same sort of identification problem that affects c1 and �1. If a consumer had I=0, usage rate was fixed and/or there were no discounting, and furthermore, if the consumer knew that he/she
49 Suppose we had not constrained c1 = 0. Higher inventory carrying costs would also cause one to buy small quantities frequently, so as to smooth inventories over time. Thus, a reduction in �0 and in increase in c1 would both lead to more frequent small purchases and hence smoother inventories. However, the former would increase overall demand, while the later would reduce it. So the likelihood would not be flat in these two parameters.
46
would not buy again until Q was used up, then there would be a locus of c2 and �2 values that would generate equal present values of fixed plus carrying costs for a purchase Q.
However, c2 and �2 are separately identified by variation in inventories and the probability of subsequent purchases. A large c2 says one should avoid buying a large quantity if current inventory is already large and/or one thinks it is likely one would want to buy again in the near future (say, because a deal is likely). In contrast, a large �2 says one should avoid buying large sizes regardless of one’s state. This will tend to make time between purchases shorter, leading to less positive duration dependence in the purchase hazard.
In contrast to fixed costs, higher inventory carrying costs should induce more positive duration dependence in the purchase hazard. A value of c2 > 0 induces one to smooth inventories, to the extent that one wishes to avoid very high inventory spikes, but it leaves one rather indifferent to fluctuations of inventories around low levels. In other words, since c2 > 0 induces a convex carrying cost function, the marginal cost of carrying inventories will be small until inventories grow quite large. Thus, purchase probability is increasing in duration since last purchase.
The simulations reported in Table A1 are consistent with these assertions. They indicate that while increases in c2 and �2 both shift down the purchase hazard, the increase in c2 makes the hazard steeper (i.e., greater positive duration dependence), while the increase in �2 makes the hazard flatter.
Next consider the role of stock out costs. The critical role of these parameters is to induce positive duration dependence in purchase probabilities. With a stock out cost, the probability of a purchase is increasing in duration since last purchase, holding price fixed. As we noted earlier, fixed costs of purchase cannot induce positive duration dependence, so there is no danger of confounding these parameters with the stock out cost parameters.50 However, an inventory carrying cost also makes purchase more likely as duration since last purchase increases. But a higher inventory carrying cost reduces demand, while a higher stock out cost increases demand, so these parameters have different effects. All these statements are verified by the simulations in Table A1.
Now consider the tastes for consumption (or utility weights) � and the usage rate R. An increase in � or R each increases demand. However, they have different effects on the duration dependence of purchase probabilities. A higher usage rate R causes important changes in the duration dependence in the purchase hazard, while a higher � does not. As the simulations in Table A1 show, the effects of increasing usage rates on the duration dependence of the purchase hazard are rather complex. For high and medium usage rate types, the increase in R leads to less positive duration dependence (i.e., the relative frequency of short inter-purchase spells increases). But for low usage rate types the hazard increases for intermediate length spells relative to both short and long spells. Table A1 also describes how key parameters affect accepted prices. It is interesting that most parameters seem to have negligible effects on accepted prices. It is also interesting that the utility weights have ambiguous effects. Of course, in a static model, an increase in the utility weights would unambiguously raise accepted prices. This is no longer true in a dynamic model, where consumers can search for good prices over time. One clear cut effect is that if we raise the
50 In addition, our fixed cost parameters (constant, linear and quadratic) are also pinned down by the relative purchase frequencies for the different sizes.
47
Heinz utility weight for type 1 consumers (i.e., the ones who strongly prefer Heinz) it raises accepted prices for Heinz. Other effects are more complex.
48
Appendix B: The Solution of the Dynamic Programming Problem
In this appendix we describe the details of how we solve the dynamic programming (DP) problem faced by households in our model. The solution of the DP problem proceeds as follows. In order to construct the value function V(Iit, I1it, Pt) in equation (15) we need to construct the objects � �� �ittititjQtR RPIIVE
it,,1, . Given those objects, the expected maximum
taken in equation (15) has simple closed form. The � �� �ittititjQtR RPIIVEit
,,1, are expectations
(over usage rate realizations) of the alternative specific value functions, which are given by equations (16) and (17). Using (16) and (17), along with (8), and letting F(R) denote the cumulative distribution function of the usage rate, we obtain: (A1) � �� �ittititjQtR RPIIVE
it,,1,
� �� �
�
�
�
�
�
�
�
�����
���
�
�
�
2321
0
)(Pr
)(1
ijtijtitijtjtitijtitit
QI
it
ijtit
ijtjit QQDQPYQIRob
RdFR
QIQI
ijtit
���
�
)(Pr)(Pr
)(21)(
21
0
2
201
ijtititijtitit
QI
itijtit
QI
itijtit
QIRobQIRob
RdFRQIcRdFRQIc ijtitijtit
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23211 ijtijtitijtjtitijtjit QQDQPYQI ���� �������
� � � �
)(Pr
)()(2
)()(2
21
22
21
2
1
ijtitit
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ijtit
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QIRob
RdFQI
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it
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it
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QIijtitit
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RdFQIRs
s ijtit��
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� �ijttitittititPR QPIIPIIVEE
tit,,1,),1,( 1111 ���
�
��
49
Note that the first term in brackets corresponds to usage rate realizations that generate no stock-out, while the second term in brackets corresponds to cases in which a stock-out does occur. The univariate integrals over Rit in (A1) can be done analytically.
The last term in (A1) is the future component, � �ijttitittititPR QPIIPIIVEEtit
,,1,),1,( 1111 ����
�
, which we must somehow compute. In our model, we assume that households solve an infinite horizon stationary problem. However, as computational device for solving the DP problem, we assume there is terminal period T at which the future component is exactly zero at all state points. Then, at t=T-1, equation (A1) takes a simple form, since the last term drops out, and we can calculate the �� �11111, ,,1,
1 �����
�
iTTiTiTTjQR RPIIVEiT
values analytically. These can then be substituted into equation (15) to obtain values for the V(Ii,T-1, I1i,T-1, Pt,T-1). Given these, it is straightforward to construct the future component terms in (A1) that are relevant for t=T-2. Given these, we can calculate the � �� �222 ,,1
��� iTTiT RP22, ,2 ��
�
iTTjQR IVEiT
I values analytically, and so on. This process of solving a finite horizon DP problem by working backward from a terminal period in which the value functions are known is called “backsolving.”
Our computational procedure for solving the infinite horizon DP problem is to backsolve the finite horizon DP problem for a sufficiently large number of time periods so that the value functions at each state point become stable, meaning that they cease to change significantly as we move further back. This approach to solving infinite horizon problems is quite common. We will make the criterion for stability more precise below.
It is important to note that the state variables in our DP problem, Iit, I1it and Pt, are continuous. Therefore, in contrast to problems with a finite number of state points, it is not possible to solve exactly for V at every state point. Thus, exact solution of the DP problem is impossible, and an approximation method must be used. We therefore introduce a polynomial approximation for V .
),1,( 111 ��� titit PII
,1,( 111 ��� titit PII )The polynomial for V is a function of total inventories, I),1,( 111 ��� titit PII it, the ratio of
quality adjusted inventories to total inventories I1it/Iit, and the vector of prices of the common size of each brand, Pjt(c) for j=1,�,4. To be precise, we specify:
� ��
�
� �
�
�
�
�
�
�
�
�
LlKk j
jmjt
lkMmm
l
it
itkitmmlktitit cP
IIICPIIVA
,...,0,...,0
4
1
)(
),()4(),...,1(
)4(),...,1(,, )]([ln1
),1,()2(
where the are parameters to be estimated, and is a set whose elements satisfy the following conditions:
)4(),..,1(,, mmlkC ),( lkM
��
�
4
1
2or 1 0)(j
,jm if , l ; 0�k 0�
��
�
4
1
1or 0)(j
jm if l and , or and ; 0� 3,2,1�k 1�l 1,0�k
��
�
4
1
0)(j
jm if l and , or l=0 and k=4, or k=0 and l=2,3,4, or k=1 and l=2. 1� 4,3,2�k
50
The structure of the set M(k,l) is set up so as to ignore various high order interaction terms, thus achieving a more parsimonious specification. For instance, the cases where the m(j) sum to 2 and k=0, l=0 correspond to squared terms in prices and interactions between each pair of prices. The requirement that k=0 and l=0 in this case means that these second order price terms are not interacted with inventories. The total number of coefficients in the approximating polynomial is 48, and the R2 is .99. The parameters are estimated by OLS regression. To obtain the sample of
data points on which the regression is run, we calculate V on
inventory/price grid points . To set up the grid, we first set the inventory grid points to be the Chebychev quadrature points on the interval from 0 to 80. The values of quality adjusted inventories and prices at each grid point are then set as follows:
)4(),..,1(,, mmlkC
),1,( ggg PII 1000�G
),1,( ggg PIIgI
First, we generate a fraction of inventories that is allotted to each brand.51 Given these fractions and the Ig, we can construct brand specific inventories , j=1,�,4. We then multiply these by the utility weights �
gjI
j to obtain the quality weighted inventory I1g. Second, the four (brand specific) prices are drawn i.i.d. from a uniform distribution on the
interval from 35 cents to 200 cents. This exceeds the range of prices for the 32 oz size observed in the data. These prices are then divided by the standard size 32 to obtain prices per ounce.
Having defined the grid over which we calculate the value functions, we can now return to the issue of convergence of the backsolving process. We backsolve until we reach a point where, in going back one additional period, the maximum percentage change in the value functions across all grid points is less than 0.1%.
We now discuss why we use I1it+1/Iit+1 as an argument in the polynomial approximation for V , rather than I1),1,( 111 ��� titit PII it+1 itself. One reason is that I1it+1 is highly collinear with Iit+1, while I1it+1/Iit+1 is not. Thus, the OLS regression we use to estimate theC is better behaved if we use the ratio.
)4(),..,1(,, mmlk
A second reason is more subtle. The ratio specification has a computational advantage that arises because I1it+1/Iit+1 does not depend on Rit. To see this, note that if
then . Thus, using (5) and (11), we have: ijtitit QIR �� itijtitit RQII ����1
� ��
�
�
�
�
�
�
�
�
�
��
ijtit
itijtitijtjitit QI
RQIQII 11 1 �
and hence:
�
�
�
�
�
�
�
�
�
�
�
�
ijtit
ijtjit
it
t
QIQI
II �11
1
1
51 To generate the inventory shares for each brand, we draw three uniform random numbers on the interval (0,1). Denote these by u1, u2 and u3. Then numerator of the share for brand 1 is set to u , that for brand 2 is set
to , that for brand 3 is set to , and that for brand 4 is set to . The denominator of the shares are set to the sum of the four numerators. This construction guarantees that the inventory shares of the four brands sum to one.
321 uu
321)1( uuu� 321 )1)(1( uuu �� )1)(1)(1( 321 uuu ���
51
Thus, I1t+1/It+1 does not depend on Rit.52 We now describe why this is advantageous. Updating (A2) by one period and substituting for Iit+1 and I1it+1/Iit+1, we obtain:
),1,()3( 111 ��� titit PIIVA
� �� ��
�
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
LlKk j
jmjt
lkMmm ijtit
ijtjititijtitmmlk cP
QIQI
RQIC,...,0,...,0
4
1
)(1
),()4(),..,1(
l
k)4(),...,1(,, )]([ln
1
�
Since Rit does not appear in the I1it+1/Iit+1 term, the expectation over Rit is simply:
),1,()4( 111 ��� tititR PIIVEAit
� ��
�
� �
�
�
�
�
���
�
�
�
�
LlKk
kitijtitR
j
jmjt
lkMmm
l
it
itmmlk RQIEcP
IIC
it
,...,0,...,0
4
1
)(1
),()4(),..,1( 1
1)4(),...,1(,, )()]([ln1
Now, for each k, the terms can be calculated analytically using a simple
quadrature procedure.
kitijtitR RQIE
it)( ��
Finally, to obtain the future component, � �ijttitittititPR QPIIPIIVEEtit
,,1,),1,( 1111 ����
, which is the last term in (A1), we must also take an expectation with respect to price realizations at t+1. We have:
),1,()5( 1111 ����
tititRp PIIVEEAitt
� ��
�
� �
�
�
�
�
�
���
�
�
�
�
LlKk j
jmjtP
kitijtitR
l
it
it
lkMmm
mmlk cPERQIEIIC
tit
,...,0,...,0
4
1
)(1
1
1
),()4(),..,1(
)4(),...,1(,, )]([ln)(1 1
where, using equations (13) and (14), the last term in (A5) can be written:
�����
�
��
�
�
�
�
�
�
4
11
)(4
1
)(1 )](ln [)(ln
1j
jtjm
jtj
jmjtP cPEcPE
tt�
�
� �
�
�
�
�
�
�
�
��� ��
jt
jm
jtl
ltjjtjj cPcP 2
)(4
1321 )]([ln)4/1()]([ln �����
The integration over realizations of the error term for prices� in equation (13) can be done analytically, since we assume these errors are normally distributed.
jt
52 Note that if , then and , so the ratio is undefined. However, the Chebychev
quadrature points that we use always have Ii,t+1>0, so this problem does not arise. ijtitit QIR �� 01 ��itI 01 1 ��itI
52
Table 1: Probability of availability of various brands and sizes.
Sizes (oz) Brands Store Brand Del Monte Heinz Hunts
14 0.8840 0.4268 1.0000 0.0 28 0.7060 0.7817 0.9975 0.0 32 0.8840 1.0000 0.9968 0.9968 40 0.0 0.5630 0.9968 0.6071 64 0.0 0.9264 0.9968 0.9968
Table 2: Mean price of the 32 oz. size (in cents)
Store Brand Del Monte Heinz Hunts Total 90.70 105.07 115.09 104.93 104.33
Table 3: Average % Difference in per oz. prices from 32 oz. size
Oz. Size Store Brand Del Monte Heinz Hunts Total
14 33.43 67.88 54.66 48.97 28 43.47 51.53 37.26 43.51 40 32.44 33.55 39.86 35.03 64 -9.93 16.00 -6.76 0.00
53
Table 4: Estimates of the Price Process Coefficients
Parameters in Logit for Probability of Price Staying Constant
Store Brand intercept 01� 1.829 (0.01890) Del Monte intercept 02� 0.6170 (0.00901) Heinz intercept 03� 0.3079 (0.00980) Hunts intercept 04� 0.7655 (0.00814) Store Brand slope coefficient 11� 1.139 (0.0460) Del Monte slope coefficient 12� 2.004 (0.0327) Heinz slope coefficient 13� 1.908 (0.0145) Hunts slope coefficient 14� 1.577 (0.0371) Square term coefficient 2� -0.1453 (0.0239)
Parameters of the Autoregressive Process for Log Price Change Store Brand Intercept 01� 0.3851 (0.00428) Del Monte intercept 02� 0.4375 (0.00415) Heinz intercept 03� 0.5068 (0.00470) Hunts intercept 04� 0.4534 (0.00455) Slope coefficient 1� 0.4473 (0.00330) Square term coefficient 2� 0.1482 (0.00516)
Variance Covariance Matrix Parameters 11� 0.00402 (3.22E-5)
12� 0.00121 (5.10E-5)
13 0.00148 (6.37E-5)
14� 0.00014 (4.53E-5)
22� 0.01189 (5.71E-5)
23� -0.00042 (9.04E-5)
24� 0.00218 (6.14E-5)
33� 0.00891 (9.63E-5)
34� -0.00050 (6.42E-5)
44� 0.00820 (4.93E-5) Note: The brand subscripts are defined as follows: Store Brand=1, Del Monte=2, Heinz=3, Hunts=4.
54
Table 5: OLS Results for Log Prices of Atypical Sizes Relative to 32oz
Constant Terms Size (oz) Store Brand Del Monte Heinz Hunts
14 1.409 1.642 1.814 28 0.971 1.338 1.191 40 1.566 1.639 1.265 64 0.445 1.143 0.614
Slope Coefficients Size (oz) Store Brand Del Monte Heinz Hunts
14 -0.090 -0.003 -0.085 28 0.419 0.197 0.324 40 -0.211 -0.059 0.228 64 0.561 0.211 0.411
Standard Errors Size (oz) Store Brand Del Monte Heinz Hunts
14 0.046 0.033 0.045 28 0.128 0.107 0.087 40 0.145 0.060 0.096 64 0.127 0.150 0.119
55
Table 6: Parameter Estimates for the Structural Model Parameter Symbol Estimate Standard Error
Utility Type 1
Store Brand Utility Weight 11� 0.0373 0.108 Del Monte Utility Weight 12� 0.4520 0.176 Heinz Utility Weight 13� 4.0860 0.135 Hunts Utility Weight 14� 1.3924 0.174
Utility Type 2
Store Brand Utility Weight 21� 0.0099 0.174 Del Monte Utility Weight 22� 2.2218 0.150 Heinz Utility Weight 23� 3.3812 0.144 Hunts Utility Weight 24� 3.2828 0.145
Utility Type 3
Store Brand Utility Weight 31� 0.1205 0.295 Del Monte Utility Weight 32� 2.7410 0.188 Heinz Utility Weight 33� 3.0087 0.212 Hunts Utility Weight 34� 1.3046 0.397
Utility Type 4
Store Brand Utility Weight 41� 3.2608 0.214 Del Monte Utility Weight 42� 0.3618 0.339 Heinz Utility Weight 43� 2.4825 0.199 Hunts Utility Weight 44� 2.8661 0.198
Type Probabilities
Utility Type 1 1� 0.5148 0.025 Utility Type 2 2� 0.3426 0.026 Utility Type 3 3� 0.0996 0.020
Other Utility Function Parameters
Precision of Utility Shocks � 0.03125 3.51E-4 Discount factor � 0.99 �
56
Table 6: Continued Parameter Symbol Estimate Standard Error
Parameters of Stockout Costs, Inventory Carrying Costs and Fixed Costs of Purchase
Stockout cost: Constant s0 11.528 1.446 Stockout Cost: Linear Term s1 0.001728 0.370 Inventory Carrying Cost: Square Term 2c 0.006236 5.16E-5 Cost of Purchase: Constant 1� 228.46 3.527 Cost of Purchase: Size 2� -4.7263 0.271 Cost of Purchase: Size2 3� 0.06119 0.0016
Usage Rate Process: Type 1 Mean 1� 3.0186 0.063 Standard Deviation 1� 0.5111 0.011
Usage Rate Process: Type 2 Mean 2� 1.5222 0.0033 Standard Deviation 2� 1.1224 0.0054
Usage Rate Process: Type 3 Mean 3� 0.5267 0.034 Standard Deviation 3� 0.5253 0.042
Usage Rate Type Persistence Type 1 11� 0.3511 0.0060 Type 2 22� 0.9958 9.53E-4 Type 3 33� 0.9101 3.95E-4 Type 4 44� 0.9049 3.41E-4
Usage Rate Types, Initial Probability Type 1 01� 0.1245 0.031 Type 2 02� 0.2770 0.101 Type 3 03� 0.2565 0.045 Family Size Effect on Usage Rate zf 0.03484 0.097
57
Table 7: Choice Frequencies in Data vs. Model Predictions Sample Choice Frequencies
Brand Size (oz) Store Del Monte Heinz Hunts
Size Total
14 0.0159 0.0049 0.0489 0.0698 28 0.0050 0.0156 0.1498 0.1752 32 0.0326 0.0904 0.3643 0.1540 0.6413 40 0.0032 0.0444 0.0060 0.0535 64 0.0029 0.0571 0.0049 0.0649
Brand Total 0.0535 0.1170 0.6646 0.1649 1.0000 Purchase probability: 0.06768
Simulated Choice Frequencies Brand
Size (oz) Store Del Monte Heinz Hunts
Size Total 14 0.0183 0.0079 0.0426 0.0688 28 0.0158 0.0145 0.1064 0.1367 32 0.0310 0.0723 0.3636 0.1431 0.6100 40 0.0123 0.0848 0.0167 0.1139 64 0.0050 0.0486 0.0169 0.0705
Brand Total 0.0651 0.1155 0.6461 0.1768 1.0000 Purchase probability: 0.06775 Stockout probability: 0.6665 Average Inventory Level: 7.5226
58
Table 8: Average Offer and Accepted Prices in Data vs. Model Predictions Mean Offer Prices - Data
Oz. Size Store Brand Del Monte Heinz Hunts 14 3.752 5.154 5.492 28 4.024 4.830 4.901 32 2.836 3.288 3.596 3.280 40 4.007 4.742 4.502 64 2.845 4.137 3.024
Mean Offer Prices – Simulation of the Model
Oz. Size Store Brand Del Monte Heinz Hunts 14 3.752 5.154 5.491 28 4.022 4.827 4.897 32 2.833 3.284 3.594 3.273 40 4.013 4.743 4.500 64 2.835 4.133 3.023
Mean Accepted Prices - Data
Oz. Size Store Brand Del Monte Heinz Hunts 14 3.747 5.078 5.535 28 4.045 4.706 4.749 32 2.760 2.996 3.509 3.114 40 4.145 4.619 4.470 64 2.580 3.909 2.993
Mean Accepted Prices – Simulation of the Model
Oz. Size Store Brand Del Monte Heinz Hunts 14 3.737 5.136 5.464 28 3.666 4.649 4.674 32 2.785 3.006 3.463 3.099 40 3.657 4.663 4.392 64 2.638 3.302 2.789
Note: The figures in the table are cents per ounce. For accepted prices, brand totals are obtained by dividing aggregate brand sales revenue by the aggregate quantity sold of the brand (i.e., purchases of larger sizes receive more weight).
59
Table 9: Brand Switching Matrix in Data vs. Model Predictions Data
Store Brand Del Monte Heinz Hunts Store Brand 0.2719 0.1338 0.4233 0.1711 DelMonte 0.0583 0.3407 0.4111 0.1898
Heinz 0.0340 0.0698 0.7895 0.1067 Hunts 0.0678 0.1576 0.4516 0.3230
Simulation of the Model
Store Brand Del Monte Heinz Hunts Store Brand 0.2363 0.0930 0.4661 0.2047 DelMonte 0.0520 0.2270 0.4850 0.2360
Heinz 0.0468 0.0834 0.7422 0.1276 Hunts 0.0780 0.1474 0.4643 0.3103
Note: The left column reports the brand bought on the previous purchase occasion. The top row indicates the brand bought on the current purchase occasion.
60
Table 10: Effects of Temporary 10% Heinz Price Decrease On Purchase Frequencies for Heinz and Other Brands
Week Heinz Hunts Del Monte Store Brand Total
1 41.30 45.28 -3.99 -1.93 -3.58 -1.81 -3.11 -1.88 25.32 28.532 -2.07 -2.31 -1.23 -1.38 -1.23 -1.29 -1.07 -1.21 -1.75 -1.963 -1.56 -1.72 -0.94 -1.02 -0.80 -0.89 -0.79 -0.88 -1.31 -1.454 -1.40 -1.54 -0.68 -0.72 -0.57 -0.63 -0.58 -0.65 -1.13 -1.245 -0.80 -0.92 -0.53 -0.55 -0.37 -0.41 -0.39 -0.43 -0.68 -0.776 -0.55 -0.58 -0.26 -0.27 -0.21 -0.21 -0.18 -0.19 -0.44 -0.467 -0.42 -0.42 -0.15 -0.19 -0.12 -0.14 -0.24 -0.26 -0.33 -0.348 -0.25 -0.24 -0.10 -0.12 -0.06 -0.07 -0.09 -0.10 -0.19 -0.199 -0.17 -0.19 -0.02 -0.04 -0.05 -0.06 -0.05 0.03 -0.12 -0.14
10 -0.30 -0.15 -0.04 -0.04 -0.08 -0.06 0.04 -0.05 -0.21 -0.1111 -0.13 -0.14 -0.04 -0.04 -0.08 -0.07 -0.03 0.04 -0.09 -0.1012 -0.21 -0.21 -0.03 -0.04 -0.02 -0.03 -0.12 -0.04 -0.15 -0.1513 -0.05 -0.04 0.01 -0.01 -0.01 -0.01 -0.01 -0.11 -0.04 -0.0314 -0.15 -0.01 0.02 0.01 0.00 0.00 0.01 -0.00 -0.10 -0.0115 0.02 0.04 0.03 0.04 0.01 0.03 0.01 0.02 0.02 0.04
Note: The table reports the effects of a temporary 10% price cut for Heinz on simulated weeklysales frequencies for Heinz and the other brands over a 15 week period. Changes are reported in percent terms. The first column (for each brand) shows the effect when expectations of futureprices are allowed to adjust. The second column (for each brand) shows the effect holding expectations of future prices fixed.
61
Table 11: Effects of Temporary 10% Heinz Price Decrease On Purchase Quantities for Heinz and Other Brands
Week Heinz Hunts Del Monte Store Brand Total
1 45.07 49.19 -4.02 -1.90 -3.64 -1.78 -3.15 -1.82 27.93 31.282 -2.14 -2.38 -1.22 -1.38 -1.13 -1.30 -1.08 -1.23 -1.81 -2.013 -1.51 -1.67 -0.91 -0.99 -0.80 -0.89 -0.81 -0.90 -1.29 -1.424 -1.61 -1.75 -0.65 -0.69 -0.56 -0.62 -0.60 -0.67 -1.26 -1.375 -0.87 -0.97 -0.54 -0.55 -0.36 -0.41 -0.41 -0.45 -0.73 -0.806 -0.57 -0.60 -0.26 -0.27 -0.20 -0.20 -0.19 -0.21 -0.46 -0.487 -0.43 -0.44 -0.14 -0.18 -0.11 -0.13 -0.24 -0.25 -0.33 -0.358 -0.26 -0.24 -0.13 -0.15 -0.06 -0.06 -0.09 -0.10 -0.20 -0.209 -0.18 -0.20 -0.02 -0.04 -0.04 -0.05 0.03 0.01 -0.13 -0.14
10 -0.44 -0.15 -0.04 -0.03 -0.08 -0.07 -0.05 -0.05 -0.31 -0.1111 -0.12 -0.13 -0.04 -0.04 -0.08 -0.03 0.03 -0.03 -0.09 -0.1012 -0.33 -0.32 -0.03 -0.04 -0.02 -0.01 -0.03 -0.04 -0.22 -0.2213 -0.06 -0.04 -0.01 -0.01 -0.01 0.00 -0.10 -0.10 -0.04 -0.0414 -0.19 -0.02 0.02 0.01 0.01 0.03 -0.01 -0.00 -0.12 -0.0115 0.02 0.05 0.03 0.04 0.02 0.04 0.01 0.02 0.02 0.04
Note: The table reports the effects of a temporary 10% price cut for Heinz on simulated weeklysales quantities for Heinz and the other brands over a 15 week period. Changes are reported in percent terms. The first column (for each brand) shows the effect when expectations of futureprices are allowed to adjust. The second column (for each brand) shows the effect holding expectations of future prices fixed.
62
Table 12: Predicted Effects of Permanent Changes in Heinz Pricing Policy
Permanent 10 % drop in Mean Offer Price of Heinz Store Brand Del Monte Heinz Hunts Total Purchase Probability
-8.078 -9.071 33.071 -10.100 18.038
Purchase Quantity
-8.075 -8.821 35.581 -10.186 19.844
Revenue
-8.052 -9.008 23.363 -10.242 13.151
Accepted Price (Mean)
-0.048 -0.205 -9.012 -0.063 -5.585
Permanent 50 % drop in the Standard Deviation of Heinz Offer Prices Store Brand Del Monte Heinz Hunts Total Purchase Probability
-6.351 -10.172 -5.278 -6.994 -6.200
Purchase Quantity
-7.522 -11.145 -6.642 -7.256 -7.295
Revenue
-7.385 -10.683 -4.221 -7.110 -5.485
Accepted Price (Mean)
0.141 0.520 2.594 0.157 1.952
Combined 2.2% Permanent drop in Mean and 50 % drop in Standard Deviation for Heinz Price Store Brand Del Monte Heinz Hunts Total Purchase Probability
-6.746 -12.157 1.052 -9.347 -2.774
Purchase Quantity
-7.340 -13.021 �0 -9.660 -3.622
Revenue
-7.238 -12.669 0.537 -9.545 -2.802
Accepted Price (Mean)
0.110 0.404 0.530 0.127 0.851
Note: The table reports the percentage changes in each of the indicated quantities for the period after the policy change, compared to a baseline simulation under the present pricing policy. The mean accepted prices are obtained by dividing aggregate sales (over all sizes) by aggregate quantity.
63
Tab
le A
1: E
ffec
t of I
ncre
asin
g Fu
ndam
enta
l Par
amet
ers o
n K
ey F
eatu
res o
f the
Dat
a
Purc
hase
Haz
ard
Pu
rcha
se F
requ
ency
Para
met
er
Gen
eral
Lev
el
Dur
atio
nD
epen
denc
eO
vera
llLa
rge
Size
vs.
Smal
l Siz
e
Mea
n A
ccep
ted
Pric
e
Car
ryin
g C
ost
Q
uadr
atic
- c
2 -
+-
-�
0
Fixe
d C
ost
of
Purc
hase
C
onst
ant -
�0
- 0
- +
� 0
Line
ar - �
1 -
0-
-�
0
Q
uadr
atic
- �
2 -
-
---
� 0
St
ock
Out
Cos
t
Con
stan
t – s 0
++
++
� 0
Line
ar –
s 1
Smal
l +
0
Smal
l +
Smal
l +
� 0
Util
ity W
eigh
ts -�
+0
+
+A
mbi
guou
s (sm
all +
ow
n ef
fect
for p
refe
rred
bra
nd)
Usa
ge R
ate
Hig
h Ty
pe
+ at
ear
ly w
eeks
onl
y --
+
++
for 6
4 oz
. Onl
y
M
ediu
m T
ype
- at
ear
ly w
eeks
(� 5
) +
at w
eeks
6+
--
�
0
+ �
0
Lo
w T
ype
+ at
inte
rmed
iate
wee
ks o
nly
+ at
ear
ly w
eeks
- a
t lat
er w
eeks
+
+�
0
Not
e: “
+” d
enot
es in
crea
se a
nd “
-“ d
enot
es d
ecre
ase.
A “
- -“
deno
tes t
hat i
ncre
asin
g a
para
met
er re
sults
in a
n ex
cept
iona
lly sh
arp
decr
ease
for s
ome
data
feat
ure,
rela
tive
to th
e m
agni
tude
of t
he p
aram
eter
’s e
ffec
t on
othe
r dat
a fe
atur
es
Figu
re 1
: Obs
erve
d Fr
eque
ncy
of T
otal
Pur
chas
es
0
0.02
0.04
0.06
0.080.1
0.12
0.14
010
2030
4050
60
tota
l num
ber o
f pur
chas
es
65
Figu
re 2
: Int
erpu
rcha
se T
ime
Dis
trib
utio
n
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Inte
rpur
chas
e sp
ell l
engt
h in
wee
ks
sim
ulat
ion
data
s
66
Figu
re 3
: Sur
vivo
r Fun
ctio
n
0
0.2
0.4
0.6
0.81
1.2
no-p
urch
ase
spel
l len
gth
in w
eeks
sim
ulat
ion
data
67
Figu
re 4
: Pur
chas
e H
azar
d
0
0.050.1
0.150.2
0.250.3
0.35
No-
purc
hase
spe
ll le
ngth
in w
eeks
sim
ulat
ion
data
68